(but %!PS not EPSF; comments have been disabled) /TeXDict 200 dict def TeXDict begin /N /def load def /B{bind def}N /S /exch load def /X{S N}B /TR /translate load N /isls false N /vsize 10 N /@rigin{ isls{[0 1 -1 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale Resolution VResolution vsize neg mul TR matrix currentmatrix dup dup 4 get round 4 exch put dup dup 5 get round 5 exch put setmatrix}N /@letter{/vsize 10 N}B /@landscape{/isls true N /vsize -1 N}B /@a4{/vsize 10.6929133858 N}B /@a3{ /vsize 15.5531 N}B /@ledger{/vsize 16 N}B /@legal{/vsize 13 N}B /@manualfeed{ statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{/nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{/sf 1 N /fntrx FMat N df-tail} B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0]N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{128 ch-data dup length 3 sub get sub}B /ch-yoff{ ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 add]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg}bind{adv 1 chg nd}bind{1 add chg}bind{1 add chg nd}bind{adv lsh}bind{ adv lsh nd}bind{adv rsh}bind{adv rsh nd}bind{1 add adv}bind{/rc X nd}bind{1 add set}bind{1 add clr}bind{adv 2 chg}bind{adv 2 chg nd}bind{pop nd}bind]N /D{ /cc X dup type /stringtype ne{]}if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto}N /eop{clear SI restore showpage userdict /eop-hook known{eop-hook}if}N /@start{userdict /start-hook known{start-hook}if /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for}N /p /show load N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval (Display)eq}{pop false}ifelse}{false}ifelse end{{gsave TR -.1 -.1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /a{moveto}B /delta 0 N /tail{dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail} B /c{-4 M}B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{ /SS save N}B /eos{clear SS restore}B end TeXDict begin 1000 300 300 @start /Fa 1 108 df<12781218A35A13C0EA3360EA3440EA 7800127E12631320EAC340EAC1800B0E7E8D10>107 D E /Fb 1 1 df<EAFFFEA20F027D8516> 0 D E /Fc 1 45 df<120E121E123FA2123E121E12021204A212081210A2122012C0080E7D850F >44 D E /Fd 4 51 df<120412081210123012201260124012C0A8124012601220123012101208 120406167D8F0B>40 D<1280124012201230121012181208120CA8120812181210123012201240 128006167E8F0B>I<121812F81218AA12FF080D7D8C0E>49 D<123EEA4180EA80C012C01200A2 EA0180EA03001204EA08401230EA7F8012FF0A0D7E8C0E>I E /Fe 22 121 df<13181330136013C01201EA0380EA0700A2120E121E121C123CA35AA412F85AAB7E1278A47E
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1520 y(where)i(on)f(the)h(righ)o(t.")e(Th)o(us,)h(it)g(is)g(not)g(a)g(go)q(o) q(d)g(idea)g(to)g(mak)o(e)e(the)j(assignmen)o(t)e(x)450 1570 y(:=)h(x)h(+)f(1.)20 b(Although)14 b(this)g(is)g(an)h(extremely)f(common)d (construction)16 b(in)e(n)o(umeric)450 1620 y(languages,)f(it)g(is)h(most)f (un)o(wise)h(in)f(a)h(sym)o(b)q(olic)e(language.)512 1671 y(Moreo)o(v)o(er,)f (Lop)q(ez's)g(Large)g(La)o(w)f(precludes)i(assigning)d(v)n(alues)h(to)g(the)h ("w)o(orking)450 1721 y(v)n(ariables")k(x,)g(y)m(,)h(z,)g(etc.)25 b(A)16 b(strict)g(distinction)g(b)q(et)o(w)o(een)h(the)f(letters)h(b)q(eing)f (used)450 1771 y(for)h("names")f(on)h(the)g(left,)h(and)f("v)n(ariables")f (on)g(the)i(righ)o(t)f(mak)o(es)f(Maple)h(more)450 1821 y(user-friendly)m(.) 512 1873 y(Since)g(w)o(e)f(ha)o(v)o(e)g(demonstrated)h(ho)o(w)e(to)i(mak)o(e) d(an)i(assignmen)o(t)f(to)h(the)h(name)450 1922 y("f")c(w)o(e)h(next)h(sho)o (w)e(ho)o(w)h(to)g("erase")g(that)g(assignmen)o(t.)512 1994 y Fn(>)582 1998 y Fo(f;)512 2069 y Fn(>)582 2073 y Fo(f)21 b(:=)h('f';)512 2144 y Fn(>)582 2148 y Fo(f;)512 2232 y Fp(First,)12 b("erasing")f(consists)h(of)f(assigning)f(f)h(to)g(its)g("letter)h(v)n(alue") f(whic)o(h)g(is)g(what)450 2282 y(the)k(single)g(quotes)g(accomplish.)k (Second,)c(en)o(tering)g("f;")e(in)o(terrogates)j(Maple)e(for)450 2332 y(what)g(it)f(kno)o(ws)h(ab)q(out)g(that)g(name.)j(Maple)c(ec)o(ho)q(es) j(the)e(con)o(ten)o(ts)h(of)e(that)h(name.)512 2384 y(It)i(is)f(p)q(ossible)h (to)f(create)i(unfathomable)c(Maple)i(W)m(orksheets.)24 b(Violations)14 b(of)450 2434 y(Lop)q(ez's)g(Large)f(La)o(w)f(can)h(lead)g(to)g(suc)o(h)g (di\016culties,)g(and)f(so)h(can)g(misunderstand-)p eop bop 150 275 a Fm(1.1)13 b(Maple)h(On)g(Line)988 b(3)150 391 y Fp(ing)11 b(Maple's)h(\014le)f(managemen)o(t.)k(F)m(or)d(example,)e(supp)q (ose)j(Lop)q(ez's)g(Large)f(La)o(w)f(has)150 441 y(b)q(een)k(violated)e(b)o (y)212 511 y Fn(>)282 515 y Fo(x)21 b(:=)h(3;)212 598 y Fp(Muc)o(h)16 b(later)f(in)g(the)h(W)m(orksheet,)g(the)f("v)n(ariable")f(x)h(is)g(used)h (sym)o(b)q(olically)m(,)c(as)150 647 y(in)212 717 y Fn(>)282 721 y Fo(f)21 b(:=)h(x*sin\(Pi*x\);)212 804 y Fp(What)12 b(happ)q(ened?)18 b(Wh)o(y)11 b(didn't)g(the)h(form)o(ula)e Fj(x)d(sin)g(\031)h(x)j Fp(app)q(ear?)18 b(Since)12 b(x)g(has)150 854 y(b)q(een)j(assigned)f(the)h(v) n(alue)e(3,)g(Maple)h(computed)f Fj(sin)q Fp(\(3)7 b Fj(\031)q Fp(\))14 b(and)g(got)f(0.)212 924 y Fn(>)282 928 y Fo(x)21 b(:=)h('x';)212 997 y Fn(>)282 1001 y Fo(f;)212 1084 y Fp(Violating)11 b(Lop)q(ez's)j(Large)f(La)o(w)f(indeed)h(has)g(consequences.)21 b(It)12 b(is)h(not)g(enough)150 1134 y(to)18 b("erase")h(x.)30 b(Once)20 b(the)e(assignmen)o(t)f(to)h(f)g(is)g(the)h(n)o(um)o(b)q(er)e(0,)i (it)e(remains)g(the)150 1184 y(n)o(um)o(b)q(er)9 b(0.)17 b(Ha)o(ving)9 b(a)g(clear)i(strategy)g(for)e(w)o(orking)g(with)h(Maple)g(prev)o(en)o(ts)h (needless)150 1234 y(frustration.)212 1285 y(The)16 b(next)g("time-b)q(om)o (b")c(is)j(more)f(subtle.)23 b(Supp)q(ose)17 b(that)e(f)g(has)h(the)g(v)n (alue)e(0)150 1335 y(from)f(the)j(ab)q(o)o(v)o(e)f(calculations.)21 b(And)16 b(supp)q(ose)g(that)f(all)f(record)j(of)d(the)i(existence)150 1385 y(of)e(f)g(in)h(the)g(W)m(orksheet)g(is)g(remo)o(v)o(ed)f(b)o(y)g (deleting)h(the)g(app)q(earance)h(of)e(f)g(from)f(the)150 1434 y(W)m(orksheet.)24 b(The)17 b(letter)f("f")f(is)h(still)f(assigned)h(the)h(v) n(alue)e(0.)23 b(In)16 b(fact,)g(it's)f(w)o(orse)150 1484 y(than)k(that.)34 b(If)19 b(another)g(blank)g(W)m(orksheet)g(is)g(op)q(ened)h(via)f(the)g(men)o (u)f(options)150 1534 y(File/New,)e(this)g(new)g(W)m(orksheet)h(will)d(still) h(ha)o(v)o(e)h(f)f(assigned)h(the)h(v)n(alue)e(0.)24 b(This)150 1584 y(is)13 b(b)q(ecause)h(b)q(oth)f(W)m(orksheets)h(share)g(the)g(same)e (memory)m(,)d(and)k(what)g(is)g(kno)o(wn)f(to)150 1634 y(Maple)j(in)g(one)h (W)m(orksheet)g(is)g(kno)o(wn)f(to)g(the)h(other.)24 b(On)16 b(some)f(platforms,)e(with)150 1683 y(the)g(righ)o(t)f(initialization)d(of)j (Maple,)f(eac)o(h)i(W)m(orksheet)g(can)f(ha)o(v)o(e)g(its)h(o)o(wn)e(attac)o (hed)150 1733 y(memory)m(.)17 b(Unless)e(y)o(ou)f(kno)o(w)g(for)h(sure)g(ho)o (w)f(y)o(our)h(cop)o(y)f(of)g(Maple)g(is)h(installed,)e(it)150 1783 y(is)i(b)q(est)i(to)e(assume)g(that)g(all)f(W)m(orksheets)j(share)f(a)f
(single)g(memory)m(,)d(and)j(alw)o(a)o(ys,)150 1833 y(under)i(all)f (circumstances,)i(remem)o(b)q(er)d(that)i(merely)f(remo)o(ving)f(the)i(app)q (earance)150 1883 y(of)c(an)g(assignmen)o(t)g(from)f(a)h(W)m(orksheet)h(do)q (es)g(not)g(remo)o(v)o(e)f(that)g(assignmen)o(t)g(from)150 1932 y(Maple's)h(memory)m(.)212 1984 y(Supp)q(ose)j(at)e(this)g(p)q(oin)o(t)g (in)g(y)o(our)g(exp)q(erimen)o(ts)g(with)g(a)g(W)m(orksheet)h(y)o(ou)f(ha)o (v)o(e)150 2034 y(excised)i(the)f(last)f(input/output)g(pair,)g(and)h(the)g (b)q(ottom)e(of)h(y)o(our)g(W)m(orksheet)h(no)150 2083 y(longer)g(con)o (tains)g("the)h(next)g(prompt.")d(Ho)o(w)i(do)g(y)o(ou)g(generate)i(a)e(new)h (prompt?)150 2133 y(Place)12 b(the)h(cursor)g(in)e(an)h(input)g(line.)17 b(Sim)o(ultaneously)9 b(press)14 b(the)e(k)o(eys)h(CTRL)e(and)150 2183 y(k)17 b(to)h(insert)g(a)f(prompt)g Fl(ab)n(ove)h Fp(the)g(cursor,)h (and)e(CTRL)g(and)g(j)h(to)f(insert)h(a)g(new)150 2233 y(prompt)13 b Fl(b)n(elow)i Fp(the)g(cursor.)21 b(If)14 b(y)o(ou)h(examine)e(the)i (Insert)h(men)o(u,)d(the)j("Execution)150 2283 y(Group")d(corresp)q(onds)j (to)e("a)f(new)i(prompt.")212 2334 y(Ha)o(ving)d(to)o(y)o(ed)i(with)f(the)h (men)o(u)e(bar,)h(observ)o(e)h(the)g(Help)f(men)o(u.)k(The)d(b)q(est)g(ad-) 150 2384 y(vice)d(a)g(Maple)g(user)h(can)f(receiv)o(e)h(is)f(to)g(b)q(egin)g (with)g(Help/Con)o(ten)o(ts.)17 b(The)12 b(resulting)150 2434 y(do)q(cumen)o(t)e(that)h(op)q(ens)g(is)f(h)o(yp)q(erlink)o(ed)h(to)f(all)f (sorts)j(of)e(information)d(ab)q(out)k(Maple,)p eop bop 450 275 a Fp(4)988 b Fm(1.1)13 b(Maple)h(On)g(Line)450 391 y Fp(and)g(it)g(is)h(left)f(to)g(the)h(user)h(to)e(na)o(vigate)g(through) g(instructions)i(on)e(the)h(in)o(terface,)450 441 y(and)f(on)f(Maple)h (itself.)512 492 y(Ho)o(w)o(ev)o(er,)e(to)g(get)g(help)g(on)g(a)f(command)e (whose)k(name)d(y)o(ou)i(kno)o(w,)f(y)o(ou)g(can)h(use)450 541 y(the)i(question)h(mark.)512 609 y Fn(>)582 613 y Fo(?subs)512 692 y Fp(Help)k(commands)d(don't)i(need)i(terminal)d(punctuation.)32 b(A)o(t)19 b(the)g(b)q(ottom)f(of)450 742 y(most)11 b(help)i(screens)i(is)e (a)f(section)h(of)f(Examples.)17 b(Lo)q(ok)12 b(there)i(\014rst.)k(Examples) 12 b(can)450 792 y(b)q(e)j(copied)f(and)f(pasted)i(bac)o(k)f(in)o(to)f(y)o (our)h(W)m(orksheet)g(for)g(exp)q(erimen)o(tation.)512 842 y(The)19 b(exercises)h(for)d(eac)o(h)i(section)f(in)g(this)g(man)o(ual)d(b)q (egin)j(with)g(the)g(instruc-)450 892 y(tion)c(to)h("restart)h(Maple.")d (This)i(means)f(to)g(issue)i(the)f Fk(restart)e Fp(command)f(whic)o(h)450 942 y(clears)17 b(all)d(v)n(ariables.)23 b(This)16 b(command)d(do)q(es)k(not) f(erase)h(an)o(ything)e(visible)g(in)g(the)450 992 y(W)m(orksheet,)g(so)g(if) f(a)h(W)m(orksheet)g(has)g(b)q(ecome)g(confused)h(and)f(en)o(tangled,)f (issuing)450 1042 y(a)g(restart)h(and)f(then)g(w)o(orking)f(from)f(the)j(top) e(do)o(wn)h(re-executing)h(all)e(the)h(en)o(tered)450 1091 y(commands,)k(is)i(a)g(w)o(a)o(y)f(of)g("b)q(eginning)g(at)h(the)g(b)q (eginning")f(and)h(retracing)g(the)450 1141 y(though)o(t)14 b(pro)q(cess)h(in)f(the)g(W)m(orksheet.)512 1208 y Fn(>)582 1212 y Fo(restart;)512 1280 y Fn(>)582 1284 y Fo(f;)512 1363 y Fp(W)m(e)g(next)g(illustrate)g(some)f(algebraic)g(simpli\014cations.)512 1430 y Fn(>)582 1434 y Fo(q)21 b(:=)h(1/x)f(+)h(1/y;)512 1502 y Fn(>)582 1506 y Fo(q1)f(:=)h(simplify\(q\);)512 1585 y Fp(Commands)9 b(in)i(Maple)g(t)o(ypically)g(tak)o(e)g(paren)o(theses)j(around)d(the)i (argumen)o(t)d(or)450 1635 y(argumen)o(ts.)22 b(The)16 b(letter)h("q")e(mak)o (es)f(a)h(handy)h(lab)q(el)f(b)q(ecause)i(it)e(is)g(easy)h(to)g(\014nd)450 1685 y(on)d(the)h(k)o(eyb)q(oard.)k(Re-assigning)12 b(a)h(new)h(v)o(ersion)g (of)e(a)h(name)g(to)g(itself)g(is)g(actually)450 1734 y(a)h(violation)d(Lop)q (ez's)k(Large)f(La)o(w,)f(and)h(should)f(b)q(e)i(a)o(v)o(oided.)i(Th)o(us,) 512 1802 y Fn(>)582 1806 y Fo(q)k(:=)h(simplify\(q\);)512 1885 y Fp(is)17 b(not)f(a)g(syn)o(tax)g(error,)i(but)f(is)f(just)h(plain)e(bad)h (w)o(orksmanship.)24 b(If)17 b(parts)g(of)450 1935 y(the)f(W)m(orksheet)g (are)h(re-executed,)h(whic)o(h)d(v)o(ersion)h(of)f(q)h(is)f(b)q(eing)h (referenced?)26 b(Is)450 1985 y(it)13 b(the)i(unsimpli\014ed)d(v)o(ersion)i (or)g(the)g(simpli\014ed)e(v)o(ersion?)18 b(So,)13 b(use)i(unique)f(names)450
2034 y(for)f(eac)o(h)h(meaningful)d(Maple)j(output)f(to)h(a)o(v)o(oid)e (confusion)h(when)h(exp)q(erimen)o(ting,)450 2084 y(since)h(that)g(usually)f (requires)i(editing,)d(c)o(hanging,)h(re-executing,)h(mo)o(ving)d(up)j(and) 450 2134 y(do)o(wn)e(throughout)h(the)g(W)m(orksheet.)19 b(If)13 b(the)i(same)d(letter)j(has)f(m)o(ultiple)d(meanings)450 2184 y(throughout)j(the)g(W)m(orksheet,)g(c)o(haos)g(results.)512 2234 y(Maple)c(crashes.)18 b(It)10 b(is)g(a)f(fact)h(of)g(life)f(that)g (Maple,)h(inexplicably)f(and)h(explicably)m(,)450 2284 y(crashes.)21 b(It)15 b(is)f(exceedingly)h(frustrating)f(to)g(ha)o(v)o(e)g(sp)q(en)o(t)i (an)e(hour)g(or)h(more)e(on)h(an)450 2334 y(assignmen)o(t)h(in)h(Maple,)g (only)g(to)g(lose)h(it)f(all)f(b)q(ecause)j(Maple)e(crashed.)27 b(There)17 b(is)450 2384 y(only)11 b(one)h(piece)h(of)f(advice)g(that)g(mak)o (es)f(sense)i(here,)g(and)f(that)g(is)g("Sa)o(v)o(e)g(early)m(,)f(and)450 2434 y(sa)o(v)o(e)g(often.")e(The)i(\014rst)h(time)d(a)h(W)m(orksheet)h(is)f (sa)o(v)o(ed)h(\(File/Sa)o(v)o(e)f(or)g(the)i("disk)o(ette")p eop bop 150 275 a Fm(1.1)13 b(Maple)h(On)g(Line)988 b(5)150 391 y Fp(icon)16 b(on)g(the)g(to)q(olbar\))g(Maple)f(prompts)h(for)f(a)h(\014le)g (name)f(and)h(a)f(destination)h(for)150 441 y(the)h(sa)o(v)o(ed)h(\014le.)27 b(Thereafter,)18 b(clic)o(king)e(the)h("disk)o(ette")h(icon)e(on)h(the)g(to)q (olbar,)g(or)150 491 y(en)o(tering)i(CONTR)o(OL)f(S)h(from)d(the)j(k)o(eyb)q (oard,)g(sa)o(v)o(es)g(w)o(ork)f(to)h(that)f(same)f(\014le.)150 541 y(Sa)o(v)o(e)d(early)f(and)h(sa)o(v)o(e)g(often.)k(That)c(advice)g (cannot)g(b)q(e)h(rep)q(eated)g(to)q(o)f(frequen)o(tly)m(.)212 593 y(Maple)19 b(is)f(b)q(oth)h(a)f(sym)o(b)q(olic)f(and)i(a)f(n)o(umeric)g (language.)31 b(Th)o(us,)20 b(there)g(is)e(a)150 643 y(di\013erence)i(b)q(et) o(w)o(een)g(1)e(and)h(1.0)e(in)h(Maple.)31 b(The)19 b(\014rst)g(is)g(the)g (exact)g(in)o(teger)g(1,)150 692 y(while)d(the)h(second)h(is)e(the)i(decimal) d(v)o(ersion)h(of)g(the)h(n)o(um)o(b)q(er)f(1.)26 b(Con)o(v)o(erting)16 b(the)150 742 y(exact)d(sym)o(b)q(olic)d(represen)o(tation)k(of)e(a)g(n)o(um) o(b)q(er)g(is)g(done)g(with)g(the)h Fk(ev)m(alf)f Fp(\(ev)n(aluate)150 792 y(\015oating)h(p)q(oin)o(t\))g(command.)212 864 y Fn(>)282 868 y Fo(q)21 b(:=)h(1/sqrt\(2\);)212 940 y Fn(>)282 944 y Fo(evalf\(q\);)212 1030 y Fp(Note)c(that)f(Maple)f(immediately)e (rationalizes)1018 1013 y Fi(1)p 1004 1020 44 2 v 1004 1025 a Fh(p)p 1031 1025 17 2 v 24 x Fi(2)1053 1030 y Fp(.)27 b(And)17 b(note)g(further)h(that)150 1085 y(Maple)c(can)g(pro)o(vide)g(man)o(y)e(more) g(than)i(the)h(default)e(10)h(digits.)212 1157 y Fn(>)282 1161 y Fo(evalf\(q,20\);)212 1247 y Fp(The)j(next)g(thing)f(useful)h(to)g(kno)o(w) f(ab)q(out)g(Maple)g(is)h(ho)o(w)f(to)g(reference)j(parts)150 1297 y(of)g(answ)o(ers)j(it)d(generates.)39 b(Consider)20 b(the)h(follo)o (wing)c(solution)i(to)h(a)g(quadratic)150 1346 y(equation.)212 1418 y Fn(>)282 1422 y Fo(q)h(:=)h(x^2)f(+)h(3*x)f(+)g(1)h(=)f(0;)212 1494 y Fn(>)282 1498 y Fo(solve\(q,x\);)212 1584 y Fp(Maple)14 b(has)g(returned)i(a)e Fl(se)n(quenc)n(e)h Fp(of)e(t)o(w)o(o)g(ro)q(ots,)h (whic)o(h)g(can)h(b)q(e)f(referenced)j(if)150 1634 y(a)d(tag)g(had)f(b)q(een) j(assigned)e(to)g(the)h Fk(solv)o(e)e Fp(command.)i(Th)o(us,)f(the)h(b)q (etter)h(w)o(orking)150 1683 y(strategy)f(is)212 1755 y Fn(>)282 1759 y Fo(q1)21 b(:=)h(solve\(q,x\);)212 1845 y Fp(No)o(w,)13 b(the)i(ro)q(ots)f(can)g(b)q(e)h(referenced)h(b)o(y)e(the)g(brac)o(k)o(et)h (notation)212 1917 y Fn(>)282 1921 y Fo(q1[1];)212 1993 y Fn(>)282 1997 y Fo(q1[2];)212 2083 y Fp(Th)o(us,)e(there)i(are)e(three)h(data)f (structures)j(Maple)c(uses)j(that)e(are)g(w)o(orth)g(under-)150 2132 y(standing.)29 b(Maple)17 b(uses)i Fl(se)n(quenc)n(es)p Fp(,)f Fl(lists)p Fp(,)f(and)h Fl(sets)p Fp(.)29 b(In)17 b(a)g(sequence)j (items)d(are)150 2182 y(separated)i(b)o(y)f(commas.)28 b(A)18 b(list)g(is)g(a)f(sequence)k(enclosed)e(b)o(y)f(square)g(brac)o(k)o(ets:)150 2232 y([a,b,c].)32 b(A)19 b(set)g(is)g(a)g(sequence)i(enclosed)f(b)o(y)f (curly)g(braces:)29 b Fg(f)p Fp(a,b,c)p Fg(g)p Fp(.)j(The)19
b(list)150 2282 y(preserv)o(es)d(order)f(and)f(replicas.)k(The)c(set)h(do)q (es)g(not.)212 2354 y Fn(>)282 2358 y Fo([a,b,a,a,c];)212 2430 y Fn(>)282 2434 y Fg(f)p Fo(a,b,a,a,c)p Fg(g)p Fo(;)p eop bop 450 275 a Fp(6)988 b Fm(1.1)13 b(Maple)h(On)g(Line)512 391 y Fp(Eac)o(h)i(structure)h(re\015ects)g(v)n(alid)d(mathematical)e(usage,) j(and)g(Maple)g(has)g(com-)450 441 y(mands)e(for)g(manipulating)e(eac)o(h)j (data)g(structure)i(prop)q(erly)m(.)512 492 y(Although)c(rep)q(etitiv)o(e)h (tasks)g(can)g(b)q(e)f(implemen)o(ted)e(in)i(Maple)g(b)o(y)g(cop)o(ying)g (and)450 541 y(pasting)20 b(input)h(lines,)g(a)g(for-lo)q(op)e(is)h(the)i (appropriate)e(w)o(a)o(y)g(to)h(rep)q(eat)g(similar)450 591 y(instructions.)30 b(In)18 b(this)f(man)o(ual,)f(the)i(for-lo)q(op)e(is)i(en) o(tered)h(as)f(a)f(single)g(input)h(as)450 641 y(follo)o(ws.)512 708 y Fn(>)582 712 y Fo(for)j(k)h(from)f(1)g(to)g(3)h(do)f(x.k)h(:=)f(k^2;)g (od;)512 791 y Fp(All)16 b(three)j(input)d(lines)h(are)h(connected)h(to)d (the)i(one)f(prompt)f(b)o(y)h(en)o(tering)g(all)450 841 y(lines)i(but)g(the)g (last)g(with)f(SHIFT)i(ENTER,)e(rather)i(than)f(simply)d(ENTER.)j(In)450 890 y(Release)13 b(4)f(this)h(is)f(no)o(w)g(more)g(aesthetic)i(than)e (practical,)h(but)f(if)g(the)h(three)h(lines)f(of)450 940 y(co)q(de)i(ab)q(o) o(v)o(e)f(are)g(en)o(tered)i(in)o(to)e(Maple)g(V)g(Release)h(4)f(with)f(just) i(the)g(ENTER)f(k)o(ey)m(,)450 990 y(the)d(\014rst)f(line)g(will)e(generate)k (a)d(complain)o(t)f(ab)q(out)i("incomplete",)f(a)g(complain)o(t)f(that)450 1040 y(disapp)q(ears)17 b(when)f(the)g(terminating)e Fk(o)q(d)i Fp(\("do")f(sp)q(elled)h(bac)o(kw)o(ards\))g(is)g(en)o(tered.)450 1090 y(In)11 b(previous)h(v)o(ersions)f(of)g(Maple,)g(failure)f(to)h(k)o(eep) h(the)g(lines)f(of)f(a)h(for-lo)q(op)f(together)450 1139 y(could)i(lead)g(to) g(terrible)g(consequences)j(if)c(c)o(hanges)i(w)o(ere)g(made)e(to)h (individual)e(lines)450 1189 y(of)j(the)i(lo)q(op.)i(In)d(Release)g(4)g(this) g(is)g(no)f(longer)h(suc)o(h)h(a)e(problem.)512 1240 y(One)18 b(adv)n(an)o(tage)e(of)g(the)i(notation)e(x1,)h(x2,)f(x3)h(is)g(that)g(suc)o (h)g(ob)r(jects)h(can)f(b)q(e)450 1289 y(referenced)f(collectiv)o(ely)e(b)o (y)512 1356 y Fn(>)582 1360 y Fo(x.\(1..3\);)512 1439 y Fp(F)m(or)19 b(large)h(collections)f(of)g(similar)e(ob)r(jects)k(this)f(turns)g(out)f(to)h (b)q(e)g(a)f(handy)450 1489 y(device)c(for)e(sa)o(ving)g(rep)q(etitiv)o(e)i (and)f(tedious)g(t)o(yping.)512 1540 y(W)m(e)g(will)f(b)q(e)i(concerned)h (primarily)c(with)i(Maple's)g(functionalit)o(y)f(in)h(linear)g(al-)450 1589 y(gebra.)35 b(Maple's)19 b(co)q(de)h(is)f(mo)q(dularized,)g(bundled)h (in)o(to)f(related)h(groups)f(called)450 1639 y(pac)o(k)n(ages.)e(There)d (are)e(some)f(32)g(pac)o(k)n(ages)h(in)g(Release)g(4,)g(all)f(of)g(whic)o(h)h (are)g(presen)o(t)450 1689 y(in)h(ev)o(ery)i(prop)q(erly)f(installed)g(v)o (ersion)g(of)f(Maple.)18 b(The)c(command)512 1756 y Fn(>)582 1760 y Fo(?packages)512 1839 y Fp(brings)e(up)h(the)f(list)g(of)f(pac)o(k)n (ages,)i(and)e(the)i(names)f(of)f(the)i(pac)o(k)n(ages)f(are)g(h)o(yp)q(er-) 450 1889 y(link)o(ed)h(to)h(more)f(information)e(ab)q(out)j(the)g(individual) e(pac)o(k)n(ages.)512 1939 y(The)20 b(pac)o(k)n(age)e(w)o(e)h(will)e(use)j (most)e(is)g(the)i Fl(linalg)e Fp(pac)o(k)n(age,)i(itself)e(con)o(taining)450 1989 y(more)10 b(that)g(100)g(commands)f(for)h(manipulating)d(v)o(ectors)12 b(and)f(matrices.)16 b(The)11 b Fl(linalg)450 2039 y Fp(pac)o(k)n(age)j(is)f ("loaded")g(in)o(to)g(Maple)h(via)f(the)i(command)512 2105 y Fn(>)582 2109 y Fo(with\(linalg\):)512 2188 y Fp(Notice)i(that)f(the)h (terminal)d(punctuation)j(here)g(is)f(the)h(colon)f(\(:\))23 b(whic)o(h)16 b(sup-)450 2238 y(presses)h(output.)k(This)14 b(pac)o(k)n(age)h(will)e(b)q(e)i(loaded)f(for)h(ev)o(ery)g(exercise)i(set,)e (and)g(w)o(e)450 2288 y(will)i(w)o(an)o(t)i(to)f(suppress)j(the)e(listing)f (of)g(the)h(more)f(than)h(100)f(commands)e(made)450 2338 y(presen)o(t)f(b)o (y)f(this)g(pac)o(k)n(age.)512 2411 y(First,)g(w)o(e)g(en)o(ter)h(the)g (matrix)d(A)i(=)1089 2352 y Ff(\024)1131 2386 y Fp(1)42 b(2)1131 2435 y(3)g(4)1235 2352 y Ff(\025)1257 2411 y Fp(.)p eop bop 150 275 a Fm(1.1)13 b(Maple)h(On)g(Line)988 b(7)212 387 y Fn(>)282 391 y Fo(A)21 b(:=)h(matrix\(2,2,[1,2,)o(3,4]\))o(;)212
468 y Fp(It)15 b(app)q(ears)h(easier)g(to)f(pro)o(vide)g(the)g Fk(matrix)f Fp(command)e(with)j(a)f(single)h(list)g(of)150 518 y(en)o(tries,)j(letting)d(Maple)i(wrap)f(them)f(according)i(to)f(the)h (dimensions)e(giv)o(en)h(\014rst.)150 568 y(The)e(alternativ)o(e)f(is)h(to)f (giv)o(e)h(the)g Fk(matrix)e Fp(command)f(a)i(list)g(of)g(lists,)g(the)i (sub-lists)150 618 y(b)q(eing)f(the)g(ro)o(ws)g(of)g(the)g(matrix.)i(Th)o (us,)212 683 y Fn(>)282 687 y Fo(matrix\([[1,2],[3)o(,4]]\))o(;)212 764 y Fp(W)m(e)10 b(will)f(use)j(Maple's)e Fk(randmatrix)e Fp(command)g(to)i(generate)i(matrices)e(at)g(ran-)150 814 y(dom.)15 b(Maple's)c(random)f(n)o(um)o(b)q(er)g(generator)i(will)e(generate)i(the)g (same)e(sequence)k(of)150 864 y(random)9 b(n)o(um)o(b)q(ers)g(eac)o(h)i(time) e(it)h(is)g(initialized)e(b)o(y)i(starting)g(\(or)h(restarting\))g(Maple.)150 914 y(The)17 b(adv)n(an)o(tage)f(here)h(is)g(that)f(results)i(are)f(then)g (repro)q(ducible,)h(ev)o(en)f(if)f(random)150 963 y(matrices)10 b(ha)o(v)o(e)h(b)q(een)h(used.)18 b(There)12 b(is)e(a)h(w)o(a)o(y)f(of)g (setting)h(the)h(starting)f(p)q(oin)o(t)f(for)g(the)150 1013 y(random)g(n)o(um)o(b)q(er)h(generator,)h(but)g(that)f(is)g(not)h(used)g(in)f (an)o(y)g(of)g(the)h(exercises.)20 b(The)150 1063 y(help)c(\014le)g(for)f Fk(rand)p Fp(,)g(the)h(random)e(n)o(um)o(b)q(er)i(generator,)g(accessed)i(b)o (y)e(?rand,)f(will)150 1113 y(men)o(tion)c(the)i(global)e(v)n(ariable)p 657 1113 13 2 v 27 w Fl(se)n(e)n(d)i Fp(whic)o(h)f(can)h(b)q(e)g(assigned)g (v)n(alues)f(\(a)h(studen)o(t's)150 1163 y(SSN?\),)j(thereb)o(y)h(making)c (unique)j(assignmen)o(ts)f(for)g(eac)o(h)h(studen)o(t.)25 b(That)15 b(is)h(not)150 1212 y(done)e(in)g(these)h(exercises.)212 1278 y Fn(>)282 1282 y Fo(B)21 b(:=)h(randmatrix\(2,2\);)212 1359 y Fp(Matrix)c(and)f(v)o(ector)i(arithmetic)d(is)i(most)e(easily)i(done)g(b)o (y)f(applying)f Fk(ev)m(alm)150 1409 y Fp(\(ev)n(aluate)e(matrix\))e(to)i (the)g(desired)h(arithmetical)d(commands.)212 1474 y Fn(>)282 1478 y Fo(2*A)21 b(+)h(3*B)f(-)g(A^2;)212 1543 y Fn(>)282 1547 y Fo(evalm\(2*A)f(+)h(3*B)g(-)h(A^2\);)212 1624 y Fp(In)16 b(the)g(\014rst)h(instance,)g(merely)e(the)h(names)f(are)h(manipulated)e(b)o (y)h(Maple.)24 b(In)150 1674 y(the)16 b(second,)g(the)g(actual)f(en)o(tries)h (of)f(the)h(matrices)e(are)i(manipulated.)k(There)d(are)150 1724 y(times)f(and)g(places)i(for)e(eac)o(h)i(approac)o(h,)f(but)g(only)f (the)h(second)h(is)f(used)g(in)g(these)150 1774 y(exercises.)212 1824 y(Next,)11 b(w)o(e)e(address)i(matrix)d(m)o(ultiplicatio)o(n,)f(a)i(pro) q(cess)j(that)d(is)h(kno)o(wn)f(in)g(math-)150 1874 y(ematics)15 b(to)g(b)q(e)h(noncomm)o(utativ)o(e.)21 b(Th)o(us,)15 b(for)h(n)o(um)o(b)q (ers,)f(2*3)g(=)g(6,)h(but)g(for)f(ma-)150 1923 y(trices,)g(A)e(B)i(rarely)f (equals)f(B)i(A.)e(Hence,)i(Maple)e(distinguishes)h(b)q(et)o(w)o(een)i(the)e (use)150 1973 y(of)h("*")f(for)h(comm)o(utativ)n(e)e(m)o(ultiplication,)f (and)j("&*")f(for)h(the)h(noncomm)o(utativ)o(e)150 2023 y(m)o(ultiplicatio)o (n)e(of)h(matrices.)24 b(In)16 b(Release)h(4)e(Maple)h(will)f(w)o(arn)h(sp)q (eci\014cally)g(that)150 2073 y(A*B)e(for)f(matrices)g(m)o(ust)f(b)q(e)i(c)o (hanged)g(to)g(A)f(&*)h(B.)f(In)g(earlier)h(v)o(ersions,)g(the)g(user)150 2123 y(had)g(to)f(b)q(e)i(prescien)o(t.)212 2188 y Fn(>)282 2192 y Fo(evalm\(A)20 b(*)i(B\);)212 2257 y Fn(>)282 2261 y Fo(evalm\(A)e(&*)i(B\);)f(evalm\(B)f(&*)h(A\);)212 2338 y Fp(Surprisingly)m (,)15 b(v)o(ectors)j(will)c(tak)o(e)j(more)e(discussion)i(than)f(matrices.)24 b(First)17 b(of)150 2411 y(all,)f(en)o(ter)j(the)e(v)o(ector)h Fk(V)f Fp(=)637 2352 y Ff(\024)679 2386 y Fg(\000)p Fp(2)696 2435 y(5)753 2352 y Ff(\025)792 2411 y Fp(with)g(the)h(follo)o(wing)c(syn)o (tax.)28 b(In)17 b(a)g(newly)p eop bop 450 275 a Fp(8)988 b Fm(1.1)13 b(Maple)h(On)g(Line)450 391 y Fp(installed)f(cop)o(y)h(of)f(Maple)h(V)g(Release)g(4)g(the)h(output)f (will)e(b)q(e)j(as)e(y)o(ou)h(see)h(b)q(elo)o(w.)512 457 y Fn(>)582 461 y Fo(V)21 b(:=)h(vector\([-2,5]\);)512 538 y Fp(Throughout)14 b(this)g(man)o(ual)d(the)k(output)f(will)e(instead)j(app)q(ear)f(as)512 604 y Fn(>)582 608 y Fo(V)21 b(:=)h(vector\([-2,5]\);)512 685
y Fp(Wh)o(y)f(the)h(di\013erence)i(and)d(ho)o(w)g(do)g(w)o(e)h(get)g(Maple)f (to)g(render)i(v)o(ectors)g(as)450 735 y("column-lik)o(e")12 b(rather)k(than)f("ro)o(w-lik)o(e"?)k(And)c(are)h(suc)o(h)f(v)o(ectors)h ("column)e(v)o(ec-)450 785 y(tors")g(or)g("ro)o(w)f(v)o(ectors"?)512 835 y(First,)18 b(no)e(matter)g(ho)o(w)g(w)o(e)h(get)g(Maple)g(to)f(prin)o(t) h(the)g(v)o(ector,)h(it)e(is)h(alw)o(a)o(ys)e(a)450 884 y(column)d(v)o (ector.)512 934 y(Second,)h(to)f(get)g(y)o(our)g(Maple)f(session)i(to)f(prin) o(t)g(the)g(v)o(ectors)h(as)f(columns,)f(en)o(ter)450 984 y(the)j(follo)o (wing)e(instructions.)512 1050 y Fn(>)582 1054 y Fo(with\(share\):)41 b(readshare\(pvac,sy)o(stem\))o(:)512 1131 y Fp(These)19 b(t)o(w)o(o)f (commands)d(cause)k(Maple)f(to)f(load,)h(from)e(its)h(Share)i(Library)m(,)e (a)450 1181 y(\014le)h(called)f Fl(pvac)i Fp(\(prin)o(t-v)o (ector-as-column\),)e(the)i(e\013ect)g(of)e(whic)o(h)h(is)g(to)g(c)o(hange) 450 1231 y(the)d(w)o(a)o(y)e(v)o(ectors)i(are)g(prin)o(ted.)k(If,)13 b(for)h(some)f(reason,)h(this)g(functionalit)o(y)f(is)g(to)h(b)q(e)450 1281 y(switc)o(hed)h(o\013,)e(en)o(ter)i(the)g(command)512 1346 y Fn(>)582 1350 y Fo(pvac)21 b(:=)g(false:)512 1416 y Fn(>)582 1420 y Fo(print\(V\);)512 1485 y Fn(>)582 1489 y Fo(pvac:=true:)512 1555 y Fn(>)582 1559 y Fo(print\(V\);)512 1636 y Fp(In)11 b(addition)e(to)i (noting)f(ho)o(w)g(to)g(turn)i(this)e(displa)o(y)g(feature)h(on)g(and)f (o\013,)h(observ)o(e)450 1686 y(that)17 b(it)f(tak)o(es)h Fk(prin)o(t)c Fp(\(or)k Fk(ev)m(alm)p Fp(\))f(to)h(get)f(Maple)h(to)f(displa)o(y)g(the)h (con)o(ten)o(ts)g(of)f(a)450 1736 y(v)o(ector)f(or)e(matrix.)512 1786 y(F)m(or)h(the)h(adv)o(en)o(turous,)g(from)d(outside)j(Maple,)f(examine) f(the)i(\014le)f(structure)j(of)450 1836 y(the)h(Maple)g(V4)f(directory)m(.) 29 b(There)19 b(is)f(a)f(sub-folder)h(called)f Fl(Shar)n(e)h Fp(in)f(whic)o(h)g(the)450 1886 y(con)o(ten)o(ts)j(of)f(the)g(Share)h (Library)f(are)g(stored.)35 b(A)19 b(further)h(sub-folder,)g Fl(System)p Fp(,)450 1935 y(con)o(tains)c(another)g(sub-folder)g(called)g Fl(Pvac)p Fp(.)25 b(In)16 b(the)h Fl(Pvac)f Fp(folder)g(there)h(is)f(a)g (\014le)450 1985 y(Pv)n(ac.mpl)p 450 2000 168 2 v -2 w(,)c(a)g(\014le)g(con)o (taining)f(the)h(co)q(de)h(for)f(the)h(displa)o(y)e(feature)i(b)q(eing)f (discussed.)450 2035 y(If)20 b(this)g(\014le)g(can)g(b)q(e)g(rendered)i(as)f (a)e(pure)i(text)g(\014le,)g(it)e(can)h(b)q(e)h(made)e(in)o(to)g(an)450 2085 y(initialization)10 b(\014le)j(so)f(that)h(the)g(co)q(de)h(will)d(load)h (automatically)d(ev)o(ery)14 b(time)d(Maple)450 2135 y(is)17 b(launc)o(hed.)27 b(The)17 b(author)g(of)f(this)h(man)o(ual)d(has)j(had)f (this)h(co)q(de)h(running)e(as)h(an)450 2184 y(initialization)9 b(\014le)j(in)f(b)q(oth)h(Release)g(3)g(and)f(Release)i(4,)e(a)h(span)g(of)f (more)g(t)o(w)o(o)g(y)o(ears.)450 2234 y(It)j(has)g(w)o(ork)o(ed)g(p)q (erfectly)h(and)f(has)g(nev)o(er)g(giv)o(en)g(an)o(y)f(trouble)h(whatso)q(ev) o(er.)512 2284 y(T)m(o)d(obtain)g(a)h(text)g(v)o(ersion)g(of)f(the)h(\014le)g (p)o(v)n(ac.mpl)p 1133 2299 162 2 v 9 w(launc)o(h)g(a)f(text)h(editor)g(suc)o (h)h(as)450 2334 y(W)m(ord,)h(etc.)21 b(F)m(rom)13 b(within)h(the)i(text)f (editor,)g(lo)q(cate)g(the)g(\014le)g(p)o(v)n(ac.mpl)p 1456 2349 V 12 w(and)g(op)q(en)450 2384 y(it.)i(P)o(erform)9 b(a)h(Sa)o(v)o(e)h (As,)g(sa)o(v)o(e)g(the)g(\014le)f(as)h("text",)g(giv)o(e)f(it)g(the)h(name)e (Maple.ini)p 1554 2399 170 2 v 9 w(\(for)450 2434 y(the)16 b(PC;)f(on)h(the)g(Macin)o(tosh,)f(use)i(MapleInit)p 1029 2448 178 2 v -1 w(,)f(and)f(for)g(UNIX,)h(use)g(.mapleinit)p 1590 2448 183 2 v -3 w(\).)p eop bop 150 275 a Fm(1.1)13 b(Maple)h(On)g(Line)988 b(9)150 391 y Fp(Be)20 b(sure)g(that)f(y)o(our)g(text)g(editor)h(do)q(es)f(not)g(add)g(a) g(hidden)g(.txt)g(or)g(other)g(suc)o(h)150 441 y(extension.)36 b(Quit)20 b(the)g(editor,)g(and)g(drop)g(the)g(initialization)d(\014le)i(in)o (to)g(the)h(Lib)150 491 y(sub-folder)d(of)f(the)h(Maple)g(V4)f(folder)h(for)f (the)i(PC,)e(drop)h(it)f(in)o(to)g(the)i(Maple)e(V4)150 541 y(folder)11 b(for)g(the)h(Mac,)g(and)f(exp)q(erimen)o(t)h(with)f(where)h(to)g (put)f(it)g(on)h(a)f(UNIX)h(system.)150 591 y(Relaunc)o(h)i(Maple)f(and)h(y)o
(ou)f(will)g(automatically)e(ha)o(v)o(e)i(launc)o(hed)h(the)h Fl(pvac)f Fp(co)q(de.)212 643 y(T)m(o)i(v)o(erify)g(that)h(a)g(Maple)f(v)o (ector)i(b)q(eha)o(v)o(es)f(as)g(a)f(column)f(v)o(ector,)j(no)f(matter)150 692 y(what)d(it)f(lo)q(oks)h(lik)o(e,)e(try)i(the)h(follo)o(wing)c(exp)q (erimen)o(ts.)212 764 y Fn(>)282 768 y Fo(print\(A,V\);)212 840 y Fn(>)282 844 y Fo(evalm\(A)20 b(&*)i(V\);)212 929 y Fp(This)17 b(is)g(exactly)g(the)g(pro)q(duct)h(y)o(ou)e(should)h(obtain)f(from)f(the)i (pro)q(duct)h(A)f Fk(V)150 979 y Fp(done)f(b)o(y)f(hand,)g(treating)g Fk(V)h Fp(as)f(a)g(column)f(v)o(ector.)23 b(No)o(w)15 b(ask)h(Maple)f(to)g (con)o(v)o(ert)150 1029 y Fk(V)f Fp(to)g(a)f(matrix.)212 1101 y Fn(>)282 1105 y Fo(VC)21 b(:=)h(convert\(V,matri)o(x\);)212 1176 y Fn(>)282 1180 y Fo(type\(V,vector\);)c(type\(VC,vector\);)212 1252 y Fn(>)282 1256 y Fo(type\(V,matrix\);)g(type\(VC,matrix\);)212 1328 y Fn(>)282 1332 y Fo(VC[1,1];)212 1404 y Fn(>)282 1408 y Fo(VC[1,2];)212 1479 y Fn(>)282 1483 y Fo(VC[2,1];)212 1569 y Fp(The)c(matrix)d(V)o(C)i(do)q(es)h(not)g(ha)o(v)o(e)f(a)g(second)h (column.)i(It)d(has)h(t)o(w)o(o)f(ro)o(ws.)k(If)c(the)150 1619 y(v)o(ector)h Fk(V)e Fp(is)h(con)o(v)o(erted)h(to)f(a)f(matrix)f(data)i (structure,)h(it)f(gets)g(con)o(v)o(erted)h(to)f(a)g(2)f(x)150 1668 y(1)i(\(column\))g(matrix.)k(Maple)c(thinks)h(of)f(the)h(v)o(ector)g Fk(V)g Fp(as)f(a)h(column)e(thing,)g(ev)o(en)150 1718 y(if)g(its)h(default)f (prin)o(t)h(st)o(yle)g(is)g(to)g(mak)o(e)e(it)i(lo)q(ok)f(lik)o(e)g(a)g(ro)o (w)h(thing.)212 1770 y(Moreo)o(v)o(er,)22 b(the)f(transp)q(ose)h(of)d Fk(V)p Fp(,)h(if)g(con)o(v)o(erted)h(to)f(a)g(matrix,)g(has)g(all)f(the)150 1820 y(c)o(haracteristics)d(of)d(a)h(1)f(x)h(2)f(\(ro)o(w\))h(matrix.)212 1892 y Fn(>)282 1896 y Fo(VR)21 b(:=)h(convert\(transpo)o(se\(V\))o(,matr)o (ix\);)212 1968 y Fn(>)282 1972 y Fo(VR[1,1];)212 2043 y Fn(>)282 2047 y Fo(VR[1,2];)212 2119 y Fn(>)282 2123 y Fo(VR[2,1];)212 2208 y Fp(The)14 b(matrix)d(VR)h(do)q(es)i(not)f(ha)o(v)o(e)f(a)h(second)h (ro)o(w.)j(Maple)c(thinks)g(of)f(the)i(trans-)150 2258 y(p)q(ose)i(of)e(V)i (as)f(a)g(ro)o(w)f(thing,)h(no)g(matter)f(ho)o(w)h(it)g(prin)o(ts)g(it.)21 b(In)15 b(fact,)g(it)g(is)g(a)g(great)150 2308 y(tragedy)e(that)g(the)h (Maple)f(programmers)e(ha)o(v)o(e)i(decided)h(that)f(the)g(default)g(output) 150 2358 y(to)h(the)g Fk(transp)q(ose)e Fp(command)f(is)212 2430 y Fn(>)282 2434 y Fo(transpose\(V\);)p eop bop 450 275 a Fp(10)967 b Fm(1.1)13 b(Maple)h(On)g(Line)512 391 y Fp(The)j(evidence)h(has)f(already)f(b)q(een)i(presen)o(ted)g(that)f (Maple)f(understands)i(the)450 441 y(transp)q(ose.)24 b(It)15 b(is)g(sad,)h(indeed,)f(that)h(so)f(\014ne)h(a)f(program)f(as)h(Maple)g (should)g(ha)o(v)o(e)450 491 y(suc)o(h)e(anomalous)e(b)q(eha)o(vior)h(when)h (displa)o(ying)e(the)i(transp)q(ose)h(of)e(a)h(v)o(ector)g Fk(V)g Fp(that)450 541 y(so)h(ob)o(viously)f(has)h(the)g(inheren)o(t)h(prop)q (erties)g(of)e(a)h(column)e(ob)r(ject.)512 591 y(Inciden)o(tally)m(,)19 b(this)f(means)g(that)h(there)h(is)f(no)g(w)o(a)o(y)m(,)f(p)q(er)i(se,)g(to)f (en)o(ter)h(a)e(ro)o(w)450 641 y(v)o(ector)f(in)o(to)f(Maple.)25 b(Y)m(ou)16 b(en)o(ter)i(a)e(column)f(v)o(ector,)i(the)g(default)f(v)o(ector) i(ob)r(ject,)450 691 y(then)e(transp)q(ose)g(the)g(column)e(v)o(ector.)23 b(And,)15 b(y)o(es,)h(y)o(ou)f(liv)o(e)f(with)h(not)g(b)q(eing)g(able)450 741 y(to)f(see)h(the)f(displa)o(y)f(of)h(the)g(transp)q(ose)h(as)f(a)g(ro)o (w-lik)o(e)f(ob)r(ject.)512 791 y(Caution:)k(It)12 b(is)g(tempting)f(to)h (sidestep)h(this)f(issue)h(of)e(ro)o(w)h(and)g(column)f(v)o(ectors)450 841 y(with)i(the)g(b)q(elief)g(that)g(instead,)g(ro)o(w)f(and)h(column)e (matrices)i(will)e(b)q(e)j(used.)k(This)13 b(is)450 891 y(not)j(a)f(go)q(o)q (d)g(idea.)24 b(There)17 b(are)f(commands)d(in)j(Maple)f(that)h(sp)q (eci\014cally)g(demand)450 941 y(v)o(ectors,)k(not)e(matrices.)30 b(F)m(or)18 b(example,)f(if)h(y)o(ou)f(had)h(de\014ned,)i(not)e(V,)g(but)h(V) o(C,)450 990 y(a)d(column)g(matrix,)f(and)h(w)o(an)o(ted)h(to)g(liv)o(e)f(y)o (our)g(Maple)h(life)f(with)g(only)g(matrices,)450 1040 y(y)o(ou'd)d(run)h (afoul)f(of)512 1107 y Fn(>)582 1111 y Fo(dotprod\(VC,VC\);)512
1190 y Fp(So,)21 b(y)o(ou)f(cannot)g(liv)o(e)f(without)h(v)o(ectors,)i(and)e (if)f(y)o(ou)h(cannot)g(liv)o(e)f(without)450 1240 y(v)o(ectors,)13 b(y)o(ou)e(m)o(ust)g(then)h(face)g(the)g(issue)h(of)e(ro)o(w)g(and)h(column)e (v)o(ectors.)18 b(Sorry)m(,)12 b(but)450 1290 y(to)i(reap)i(the)f(b)q (ene\014ts)h(of)e(Maple)h(y)o(ou)f(ha)o(v)o(e)g(to)h(put)g(up)f(with)h(a)f (few)h(quirks.)20 b(Kind)450 1340 y(of)13 b(lik)o(e)g(life)g(in)h(general.) 512 1390 y(There)d(is)f(one)g(\014nal)f(issue)h(to)g(face)g(ab)q(out)g (linear)f(algebra)g(in)g(Maple.)17 b(T)m(o)9 b(p)q(erform)450 1440 y(op)q(erations)20 b(on)f(v)o(ectors)i(one)e(m)o(ust)g(map)f(the)i(op)q (erator)g(on)o(to)f(the)h(v)o(ector.)35 b(F)m(or)450 1490 y(example,)12 b(to)i(simplify)d(a)i(v)o(ector,)i(use)f(the)h(follo)o(wing)c(syn)o(tax.)512 1557 y Fn(>)582 1561 y Fo(V)21 b(:=)h(vector\([1/x+1/y,)c(1/x-1/y]\);)512 1628 y Fn(>)582 1632 y Fo(simplify\(V\);)512 1711 y Fp(Ob)o(viously)m(,)12 b(not)i(the)h(righ)o(t)e(syn)o(tax.)512 1778 y Fn(>)582 1782 y Fo(map\(simplify,V\);)512 1862 y Fp(The)i(op)q(erator)f Fk(simplify)d Fp(has)j(to)f(b)q(e)i(mapp)q(ed)e(on)o(to)g(the)i(v)o(ector.)512 1912 y(As)f(another)f(example,)f(if)g Fk(V)h Fp(is)g(a)g(function)g(of)f(t)i (and)f(y)o(ou)f(w)o(an)o(t)h(its)g(deriv)n(ativ)o(e,)450 1962 y(y)o(ou)g(use)i(the)g(follo)o(wing)c(syn)o(tax.)512 2029 y Fn(>)582 2033 y Fo(V)21 b(:=)h(vector\([sin\(t\),c)o(os\(t\))o(]\);)512 2100 y Fn(>)582 2104 y Fo(diff\(V,t\);)512 2183 y Fp(Ob)o(viously)m(,)12 b(the)j(wrong)f(syn)o(tax.)512 2250 y Fn(>)582 2254 y Fo(map\(diff,V,t\);)512 2333 y Fp(Additional)f(parameters)h(to)f(the)i(mapp)q(ed)e(op)q(erator)h(go)g (at)f(the)i(end.)512 2384 y(Unfortunately)m(,)i(there)h(are)f(t)o(w)o(o)f (exceptions)i(to)e(the)i(rule)f("Map)f(things)g(on)o(to)450 2434 y(v)o(ectors)f(and)f(matrices.")p eop bop 150 275 a Fm(1.2)13 b(Maple)h(On)g(Line)967 b(11)212 387 y Fn(>)282 391 y Fo(V)21 b(:=)h(vector\([Pi,2*Pi])o(\);)212 470 y Fn(>)282 474 y Fo(evalf\(V\);)212 553 y Fn(>)282 557 y Fo(map\(evalf,V\);)212 651 y Fp(W)m(ell,)14 b(that)i(seems)g(to)f(w)o(ork.) 23 b(Wh)o(y)14 b(raise)i(that)g(as)f(an)h(exception?)23 b(The)16 b(com-)150 701 y(mand)10 b(ev)n(alf)h(can)h(tak)o(e)g(an)g(in)o(teger)h(as)f (a)f(second)i(argumen)o(t,)e(c)o(hanging)g(the)i(n)o(um)o(b)q(er)150 751 y(of)g(digits)h(returned.)212 829 y Fn(>)282 833 y Fo(evalf\(Pi,20\);)212 928 y Fp(But)h(if)e(y)o(ou)g(try)h(that)g(for)g(the)g(v)o(ector)h(V,)e(it)h (fails.)212 1006 y Fn(>)282 1010 y Fo(map\(evalf,V,20\);)212 1105 y Fp(Nonsense)i(is)e(returned.)19 b(The)c(metho)q(d)e(that)h(w)o(orks)g (is)212 1184 y Fn(>)282 1188 y Fo(evalf\(op\(V\),20\);)212 1282 y Fp(A)g(second)h(exception)g(is)f(substitution.)212 1361 y Fn(>)282 1365 y Fo(V)21 b(:=)h(vector\([x,x^2]\);)212 1443 y Fn(>)282 1447 y Fo(subs\(x=1,V\);)212 1526 y Fn(>)282 1530 y Fo(map\(subs,V,x=1\);)212 1625 y Fp(The)15 b(syn)o(tax)e(that)h(w)o(orks)g (is)212 1703 y Fn(>)282 1707 y Fo(subs\(x=1,op\(V\)\);)212 1802 y Fp(Hence,)g(the)e(rule)g(is)g("Map)f(all)g(op)q(erators)h(except)h Fk(subs)e Fp(and)h Fk(ev)m(alf)p Fp(.)k(F)m(or)c(those,)150 1852 y(don't)i Fk(map)p Fp(,)g(but)h(use)g Fk(op)f Fp(around)h(the)g(v)o (ector.)22 b(If)14 b(y)o(ou)g(use)i Fk(map)p Fp(,)d(don't)i(use)g Fk(op)p Fp(.)150 1901 y(When)f Fk(op)f Fp(is)h(needed,)h(y)o(ou)e(don't)h (use)g Fk(map)p Fp(.")150 2143 y Fq(1.2)66 b(On)22 b(Line)150 2242 y Fp(The)17 b(exercises)h(of)e(this)g(section)h(explore)g(the)f(concept) i(of)e("span")g(b)o(y)g(visualizing)150 2292 y(randomly)h(generated)j(mem)o (b)q(ers)e(of)g(the)i(span)f(of)g(a)f(set)i(of)f(v)o(ectors.)34 b(Begin)19 b(b)o(y)150 2342 y(loading)12 b(b)q(oth)i(the)h Fl(linalg)e Fp(and)h Fl(plots)g Fp(pac)o(k)n(ages.)212 2420 y Fn(>)282 2424 y Fo(with\(linalg\):)41 b(with\(plots\):)p eop bop 450 275 a Fp(12)967 b Fm(1.2)13 b(Maple)h(On)g(Line)450 391 y Fe(Exercise)j(1)450 470 y Fp(En)o(ter)e(in)o(to)f(Maple)g(the)g(follo)o (wing)e(four)i(p)q(oin)o(ts.)19 b(F)m(or)13 b(most)g(purp)q(oses,)j(p)q(oin)o
(ts)e(can)450 520 y(b)q(e)h(represen)o(ted)h(as)e(lists,)f(a)h(data)g (structure)i(denoted)e(b)o(y)g(square)h(brac)o(k)o(ets.)512 589 y Fn(>)582 593 y Fo(P1:=[1,1];)20 b(P2:=[1,-1];)f(P3:=[-1,1];)h (P4:=[-1,-1];)512 674 y Fp(The)13 b Fl(plots)g Fp(pac)o(k)n(age)f(mak)o(es)f (a)o(v)n(ailable)f(a)j Fk(p)q(oin)o(tpl)o(ot)c Fp(command)h(that)i(will)g (plot)450 724 y(a)j(list)f(of)g(p)q(oin)o(ts.)21 b(There)16 b(are)f(a)f(n)o(um)o(b)q(er)g(of)h(options)f(to)h(this)g(command)d(that)i (will)450 774 y(v)n(ary)h(the)h(lo)q(ok)e(of)h(the)h(graph,)f(and)g(b)o(y)g (using)g(the)h(to)q(olbars)f(asso)q(ciated)h(with)f(the)450 824 y(graph,)e(man)o(y)e(c)o(haracteristics)k(of)d(the)i(plot)f(can)g(b)q(e)h (adjusted.)k(T)m(o)13 b(obtain)f(help)h(on)450 873 y(this)h(command,)d(en)o (ter)512 942 y Fn(>)582 946 y Fo(?pointplot)512 1027 y Fp(T)m(o)i(plot)h (these)h(p)q(oin)o(ts,)e(en)o(ter)512 1096 y Fn(>)582 1100 y Fo(pointplot\([P.\(1.)o(.4\)]\))o(;)450 1223 y Fe(Exercise)k(2)450 1316 y Fp(En)o(ter)e(the)f(v)o(ectors)h Fk(A)f Fp(=)873 1257 y Ff(\024)916 1291 y Fp(1)916 1340 y(1)958 1257 y Ff(\025)993 1316 y Fp(and)g Fk(B)g Fp(=)1168 1257 y Ff(\024)1211 1291 y Fp(2)1211 1340 y(3)1252 1257 y Ff(\025)1274 1316 y Fp(.)512 1415 y Fn(>)582 1419 y Fo(A:=vector\([1,1]\))o(;)19 b(B:=vector\([2,3]\);)512 1500 y Fp(Using)14 b(the)h Fk(p)q(oin)o(tpl)o(ot)c Fp(command,)f(plot)k (these)h(t)o(w)o(o)f(v)o(ectors)h(as)f(p)q(oin)o(ts)g(in)f(the)450 1550 y(plane.)25 b(Note)17 b(the)g(additional)d(parameter)i Fl(view)p Fp(,)g(whic)o(h)h(sets)g(a)f(viewing)g(windo)o(w)450 1600 y(for)e(the)g(graph.)512 1669 y Fn(>)582 1673 y Fo(pointplot\([A,B],)o (view=)o([0..3)o(,0..3)o(]\);)512 1754 y Fp(a\))f(Pro)q(duce)h(sev)o(eral)f (di\013eren)o(t)h(linear)f(com)o(binations)d(of)i Fk(A)h Fp(and)g Fk(B)p Fp(.)f(Appropri-)450 1804 y(ate)i(syn)o(tax)g(for)g(doing)f(this)h(w)o (ould)f(b)q(e)h(as)g(follo)o(ws.)512 1872 y Fn(>)582 1876 y Fo(evalm\(2*A)20 b(+)h(3*B\);)512 1958 y Fp(b\))13 b(Use)h(Maple's)f(random)f (n)o(um)o(b)q(er)g(generator)i(to)f(create)h(sev)o(eral)g(random)d(lin-)450 2008 y(ear)19 b(com)o(binations)d(of)i(A)h(and)f(B.)g(First,)i(de\014ne)f(a)g (function)f(f)g(whic)o(h)g(generates)450 2057 y(random)10 b(four-digit)g(n)o (um)o(b)q(ers)i(in)f(the)h(in)o(terv)n(al)f(\(-1,1\).)17 b(This)11 b(is)h(done)g(with)f(Maple's)450 2107 y Fk(rand)j Fp(function)i(as)f(follo)o (ws.)22 b(The)16 b Fk(ev)m(alf)f Fp(command)e(c)o(hanges)j(exact)h(fractions) e(to)450 2157 y(decimals.)512 2226 y Fn(>)582 2230 y Fo(f:=evalf\(rand\(-1)o (0000.)o(.1000)o(0\)/10)o(000\):)512 2311 y Fp(Then,)f(in)o(v)o(ok)o(e)f(the) i(function)f(f)f(with)h(the)h(syn)o(tax)f(f\(\).)k(F)m(or)c(example,)e (create)j Fk(c)p Fp(,)450 2361 y(one)f(random)e(linear)i(com)o(bination,)d (with)i(the)i(syn)o(tax)512 2430 y Fn(>)582 2434 y Fo(c:=evalm\(f\(\)*A+f)o (\(\)*B\))o(;)p eop bop 150 275 a Fm(1.2)13 b(Maple)h(On)g(Line)967 b(13)212 391 y Fp(c\))16 b(Plot)e(enough)h(p)q(oin)o(ts)g(in)f(the)h(span)g(of)g(A)g(and)f (B)h(to)g(get)g(a)g(discernible)g(geo-)150 441 y(metric)f(\014gure.)21 b(Begin)15 b(with)g(100)f(random)f(linear)h(com)o(binations,)f(and,)h(using)g (the)150 491 y Fk(p)q(oin)o(tplo)o(t)e Fp(command,)f(plot)j(them)g(as)h(p)q (oin)o(ts)f(in)g(the)i(plane.)k(Maple's)14 b Fk(seq)g Fp(com-)150 541 y(mand)c(will)h(pro)q(duce)i(a)f(sequence)i(of)e(similar)d(ob)r(jects)k (from)e(a)g(pattern)i(pro)o(vided)f(to)150 591 y(it.)17 b(T)m(erminate)11 b(the)i(command)d(with)i(a)g(colon)g(\(:\))18 b(to)12 b(suppress)i(the)f (output.)18 b(Then,)150 640 y(feed)g(the)h(resulting)e(sequence)j(to)e(the)g Fk(p)q(oin)o(tplo)o(t)d Fp(command,)g(remem)o(b)q(ering)h(to)150 690 y(enclose)h(the)g(sequence)h(in)e(square)h(brac)o(k)o(ets)g(since)g Fk(p)q(oin)o(tplo)o(t)c Fp(requires)k(a)f Fl(list)g Fp(of)150 740 y(p)q(oin)o(ts)e(\(or)g(v)o(ectors\).)212 810 y Fn(>)282 814 y Fo(q:=seq\(evalm\(f\(\))o(*A+f\()o(\)*B\),)o(k=1..)o(100\):)212 885 y Fn(>)282 889 y Fo(pointplot\([q]\);)212 972 y Fp(d\))21 b(The)g(plot)f(in)g(part)h(\(c\))g(is)f(only)g(part)h(of)f(the)h(span,)h(the) f(p)q(ortion)f(b)q(eing)150 1022 y(determined)d(b)o(y)f(our)h(use)h(of)e
(random)f(n)o(um)o(b)q(ers)i(in)f(the)i(in)o(terv)n(al)e(\(-1,1\).)26 b(Rather)150 1072 y(than)17 b(rede\014ne)i(the)f(function)f(f,)g(try)h(m)o (ultiplyi)o(ng)c(the)k(v)o(ector)g Fk(A)f Fp(b)o(y)g(2,)h(creating)150 1122 y(another)c(plot)e(of)h(at)g(least)g(200)g(random)f(linear)g(com)o (binations)f(with)i Fk(A)h Fp(m)o(ultipli)o(ed)150 1172 y(b)o(y)g(2.)150 1297 y Fe(Exercise)j(3)150 1377 y Fp(Describ)q(e)d(in)e(w)o(ords)g(the)h(set) h(of)e(p)q(oin)o(ts)g(corresp)q(onding)h(to)g(the)g(collection)f(of)f(linear) 150 1427 y(com)o(binations)e(de\014ned)j(b)o(y)f(the)h(sum)e Fl(s)h Fk(A)g Fp(+)h Fl(t)f Fk(B)p Fp(,)f(where)j(b)q(oth)e Fl(s)g Fp(and)g Fl(t)g Fp(lie)g(in)f(closed)150 1477 y(in)o(terv)n(als)j(of)g (the)h(form)e([-2,2].)k(Plot)d(the)h(resulting)g(set)g(of)f(linear)g(com)o (binations)f(as)150 1526 y(p)q(oin)o(ts)17 b(in)f(the)h(plane,)g(using)g(a)f (di\013eren)o(t)i(color)e(than)h(used)h(in)e(Exercise)j(2.)26 b(\(See)150 1576 y(the)14 b(online)g(help)g(for)f(ho)o(w)h(to)f(sp)q(ecify)i (color)e(in)h(the)g(p)q(oin)o(tplot)f(command.\))150 1702 y Fe(Exercise)k(4)150 1781 y Fp(In)10 b(Exercise)i(9)e(of)g(the)h(non-computer) e(problems,)h(it)g(w)o(as)g(stated)h(that)g(eac)o(h)f(elemen)o(t)150 1795 y Ff(2)150 1870 y(4)198 1829 y Fj(x)199 1878 y(y)200 1928 y(z)243 1795 y Ff(3)243 1870 y(5)284 1879 y Fp(in)k(the)g(span)g(of)212 2034 y Fk(X)g Fp(=)308 1951 y Ff(2)308 2025 y(4)357 1984 y Fg(\000)p Fp(1)373 2034 y(1)357 2084 y Fg(\000)p Fp(1)431 1951 y Ff(3)431 2025 y(5)472 2034 y Fp(and)g Fk(Y)g Fp(=)649 1951 y Ff(2)649 2025 y(4)697 1984 y Fg(\000)p Fp(1)713 2034 y(3)713 2084 y(2)771 1951 y Ff(3)771 2025 y(5)813 2034 y Fp(satis\014es)g(5)7 b Fj(x)i Fp(+)h(3)d Fj(y)j Fg(\000)g Fp(2)d Fj(z)13 b Fp(=)f(0.)212 2135 y(a\))g(W)m(rite)f Fk(C)h Fp(=)g(a)g Fk(X)g Fp(+)g(b)f Fk(Y)p Fp(,)h(the)g(expression)h(for)e(the)i(general)f(linear)f(com)o(bina-) 150 2185 y(tion)f(of)h(A)g(and)f(Y.)h(Then)g(sho)o(w)g(that)g(the)g(comp)q (onen)o(ts)g(of)f(the)i(v)o(ector)f Fk(C)h Fp(satisfy)e(the)150 2235 y(equation)i(of)f(the)h(plane)g(declared)h(ab)q(o)o(v)o(e.)k(This)12 b(is)f(most)g(e\013ectiv)o(ely)i(accomplished)150 2285 y(as)h(follo)o(ws.)212 2355 y Fn(>)282 2359 y Fo(X)21 b(:=)h(vector\([-1,1,-1])o(\);)d(Y)i(:=)h (vector\([-1,3,2]\))o(;)212 2430 y Fn(>)282 2434 y Fo(C)f(:=)h(evalm\(a*X)e (+)h(b*Y\);)p eop bop 450 275 a Fp(14)967 b Fm(1.3)13 b(Maple)h(On)g(Line)512 387 y Fn(>)582 391 y Fo(q)21 b(:=)h(5*x)f(+)h(3*y)f(-)g(2*z)g(=)h(0;)512 460 y Fn(>)582 464 y Fo(q1)f(:=)h(subs\(x=C[1],)d(y=C[2],)h(z=C[3],)h(q\);) 512 546 y Fp(b\))e(Plot,)f(as)h(p)q(oin)o(ts)f(in)g Fj(R)947 531 y Fi(3)965 546 y Fp(,)h(a)f(few)g(h)o(undred)h(elemen)o(ts)g(of)e(this)i (span.)31 b(This)450 596 y(requires)11 b(use)f(of)e(Maple's)h Fk(p)q(oin)o(tplot)o(3d)d Fp(command,)h(the)j(3d)f(analog)f(of)h Fk(p)q(oin)o(tpl)o(ot)o Fp(.)450 646 y(Begin)15 b(b)o(y)f(forming,)e(via)h Fk(seq)p Fp(,)h(a)g(sequence)j(of)d(random)f(v)o(ectors)i(in)f(the)h(span)g (of)f(X)450 696 y(and)g(Y.)g(The)h(function)f(f)g(de\014ned)i(for)e(Problem)f (1)i(can)f(again)f(b)q(e)i(used)h(to)e(pro)o(vide)450 745 y(the)g(random)f (co)q(e\016cien)o(ts.)512 815 y Fn(>)582 819 y Fo(q:=seq\(evalm\(f\(\))o (*X+f\()o(\)*Y\),)o(k=1..)o(200\):)512 900 y Fp(The)h(syn)o(tax)g(for)g Fk(p)q(oin)o(tpl)o(ot3)o(d)d Fp(is)i(similar)f(to)h(that)h(of)f Fk(p)q(oin)o(tplo)o(t)p Fp(.)i(Ho)o(w)o(ev)o(er,)450 950 y(there)e(are)e(sev) o(eral)h(additional)d(parameters)j(whose)g(use)g(mak)o(es)e(for)h(a)g(b)q (etter)i(graph.)450 1000 y(Since)e(all)f(3d)g(plot)h(pac)o(k)n(ages)g (attempt)f(to)g(plot)g(a)h(3d)g(ob)r(ject)g(on)g(a)f(2d)h(sheet)h(of)e(pap)q (er)450 1050 y(\(or)h(computer)g(screen\),)j(there)e(is)f(a)g(need)h(for)f(a) g(reference)j(frame)c(that)h(pro)o(vides)g(the)450 1100 y(sense)17 b(of)d(depth.)23 b(T)m(ry)14 b(putting)h(a)g(b)q(o)o(x)g(around)g(y)o(our)f (graph,)h(either)h(in)o(teractiv)o(ely)450 1149 y(via)f(the)i(to)q(olbars,)f (or)g(via)g(options)g(to)g(the)h(plot)e(command)f(itself.)24 b(Ha)o(ving)16 b(clear)450 1199 y(and)e(highly)g(visible)f(lab)q(els)i(on)f (the)h(axes)g(is)f(equally)g(useful.)20 b(Th)o(us,)14 b(the)i(follo)o(wing)
450 1249 y(syn)o(tax)e(could)g(b)q(e)g(used)h(to)f(plot)f(the)h(v)o(ectors)h (randomly)d(generated)j(ab)q(o)o(v)o(e.)512 1318 y Fn(>)582 1322 y Fo(pointplot3d\([q],)j(color=black,)i(axes=boxed,)f(labels=[x,y,z],) 512 1364 y(labelfont=[TIMES,BO)o(LD,14)o(]\);)512 1446 y Fp(Observ)o(e)d (that)e(in)g(Maple,)f(3d)h(graphs)g(can)g(b)q(e)h(rotated)g(on)e(the)i (screen)h(b)o(y)e(ma-)450 1496 y(nipulation)e(with)g(the)i(mouse.)j(Clic)o(k) c(on)g(the)g(plot)g(to)g(mak)o(e)f(it)g("liv)o(e.")g(Then,)h(clic)o(k)450 1546 y(and)18 b(hold)g(do)o(wn)h(the)g(mouse)f(button,)h(dragging)f(the)h(b)q (ounding)f(b)q(o)o(x)h(that)f(no)o(w)450 1596 y(replaces)f(the)f(graph.)23 b(This)15 b(b)q(ounding)g(b)q(o)o(x)h(is)f(rotated)h(as)g(the)g(mouse)f(is)g (mo)o(v)o(ed.)450 1646 y(When)g(released,)h(the)f(b)q(ounding)f(b)q(o)o(x)h (has)g(a)f(new)i(orien)o(tation,)d(and)i(the)g(graph)g(is)450 1695 y(redra)o(wn)g(b)o(y)e(clic)o(king)g(the)i Fk(R)e Fp(\(redra)o(w\))i(on) f(the)g(to)q(olbar.)512 1747 y(T)m(ry)g(to)h(rotate)g(y)o(our)f(graph)g(to)g (demonstrate)h(that)f(the)h(plotted)g(p)q(oin)o(ts)f(lie)g(on)450 1796 y(a)g(plane.)450 1941 y Fq(1.3)89 b(On)22 b(Line)450 2034 y Fp(Maple)14 b(Release)g(4)f(p)q(ermits)h(more)f(than)g(one)h(w)o(orksheet)h (to)f(b)q(e)h(op)q(en)f(at)f(the)i(same)450 2084 y(time.)31 b(On)19 b(some)e(platforms)g(Maple)h(can)h(b)q(e)g(put)g(in)o(to)f(the)h("m)o (ultiple)d(k)o(ernels")450 2133 y(mo)q(de)d(in)h(whic)o(h)g(eac)o(h)h(w)o (orksheet)g(is)f(attac)o(hed)h(to)f(its)h(o)o(wn)e("k)o(ernel,")h(or)g (memory)450 2183 y(state.)19 b(In)13 b(this)h(mo)q(de,)e(v)n(ariables)g (declared)j(in)e(one)g(w)o(orksheet)i(will)d(not)h(b)q(e)h(kno)o(wn)450 2233 y(to)g(an)o(y)f(other)i(w)o(orksheet)g(op)q(ened)g(sim)o(ultaneously)m (.)512 2284 y(Ho)o(w)o(ev)o(er,)k(the)g(default)f(setting)h(for)e(Maple)h (migh)o(t)e(b)q(e)j(the)g("shared)g(k)o(ernel")450 2334 y(mo)q(de)14 b(in)h(whic)o(h)h(all)e(op)q(en)i(w)o(orksheets)g(share)h(the)f(same)e (memory)f(state.)23 b(In)16 b(this)450 2384 y(case,)c(v)n(ariables)e (declared)i(in)e(one)i(w)o(orksheet)g(ha)o(v)o(e)e(the)i(same)e(v)n(alue)g (in)h(ev)o(ery)g(other)450 2434 y(w)o(orksheet)18 b(op)q(ened)f(sim)o (ultaneously)m(.)23 b(The)17 b(p)q(oten)o(tial)f(here)i(for)e(gra)o(v)o(e)g (confusion)p eop bop 150 275 a Fm(1.3)13 b(Maple)h(On)g(Line)967 b(15)150 391 y Fp(is)14 b(v)o(ery)h(high.)20 b(Since,)15 b(in)f(Release)h(3,)f(only)g(one) h(w)o(orksheet)g(could)g(b)q(e)g(attac)o(hed)g(to)150 441 y(a)f(k)o(ernel,)g (this)g(con\015ict)g(b)q(et)o(w)o(een)i(m)o(ultiple)c(w)o(orksheets)j(nev)o (er)g(arose.)k(In)14 b(Release)150 491 y(4)h(it)g(is)g(essen)o(tial)g(that)g (this)h("gotc)o(ha")e(b)q(e)i(understo)q(o)q(d.)23 b(A)15 b(simple)f (precaution)h(in)150 541 y(the)g(shared)h(k)o(ernel)f(w)o(orld)f(is)g (starting)h(ev)o(ery)g(new)g(w)o(orksheet)h(with)e(a)h Fk(restart)p Fp(,)d(a)150 591 y(Maple)i(command)d(whic)o(h)j(clears)g(memory)e(and)h (resets)j(all)d(v)n(ariables.)212 641 y(The)d(use)h(of)e(the)h Fk(restart)e Fp(command)f(at)j(the)g(b)q(eginning)f(of)g(eac)o(h)i(new)f(w)o (orksheet)150 691 y(is)k(highly)e(recommended.)212 741 y(Here,)j(w)o(e)f(b)q (oth)g(restart)h(Maple)f(and)g(load)f(the)h Fl(linalg)g Fp(pac)o(k)n(age.)212 808 y Fn(>)282 812 y Fo(restart;)212 879 y Fn(>)282 883 y Fo(with\(linalg\):) 150 1002 y Fe(Exercise)j(1)150 1119 y Fp(Obtain)c Fk(X)g Fp(=)385 1036 y Ff(2)385 1110 y(4)433 1069 y Fp(4)433 1119 y(0)433 1169 y(0)474 1036 y Ff(3)474 1110 y(5)510 1119 y Fp(+)8 b Fj(s)583 1036 y Ff(2)583 1110 y(4)632 1069 y Fg(\000)p Fp(2)648 1119 y(1)648 1169 y(0)705 1036 y Ff(3)705 1110 y(5)741 1119 y Fp(+)g Fj(t)810 1036 y Ff(2)810 1110 y(4)858 1069 y Fg(\000)p Fp(3)874 1119 y(0)874 1169 y(1)932 1036 y Ff(3)932 1110 y(5)973 1119 y Fp(as)13 b(the)h(general)f(solution)f(to)h(the)150 1215 y(equation)h Fj(x)8 b Fp(+)i(2)d Fj(y)k Fp(+)e(3)e Fj(z)14 b Fp(=)d(4.)212 1266 y(Maple's)16 b Fk(linsolv)o(e)c Fp(command,)h(the)k(general)f(linear)f (system)g(solv)o(er,)h(will)f(giv)o(e)150 1316 y(the)d(general)f(solution)f (to)h(systems)g(of)g(equations.)17 b(W)m(e)11 b(b)q(egin)g(b)o(y)g(creating)g (a)g(matrix)150 1365 y(\(the)g(co)q(e\016cien)o(t)h(matrix\))d(whose)i
(\(i,j\))e(en)o(try)j(is)e(the)h(co)q(e\016cien)o(t)h(of)e(the)h(j)f(th)h(v)n (ariable)150 1415 y(in)f(the)h(i)f(th)h(equation.)16 b(Y)m(ou)10 b(can)h(either)g(t)o(yp)q(e)g(in)f(the)h(co)q(e\016cien)o(t)g(matrix)e(A)i (directly)m(,)150 1465 y(or)17 b(use)g(the)h Fk(genmatrix)c Fp(command,)h(giving)g(a)i(list)f(of)g(equations,)h(and)g(a)f(list)h(of)150 1515 y(v)n(ariables)c(as)i(parameters.)j(Th)o(us,)c(for)g(the)h(preceding)g (linear)f(equation,)f(w)o(e)h(could)150 1565 y(en)o(ter)212 1631 y Fn(>)282 1635 y Fo(q)21 b(:=)h(x+2*y+3*z)e(=)h(4;)212 1702 y Fn(>)282 1706 y Fo(A)g(:=)h(genmatrix\([q],[x)o(,y,z])o(\);)212 1785 y Fp(This)12 b(extracts)h(the)g(co)q(e\016cien)o(t)f(matrix)e(for)i(the) g(system.)17 b(Next,)c(the)f(co)q(e\016cien)o(t)150 1835 y(matrix)e(and)i (the)h(constan)o(ts)f(on)g(the)h(righ)o(t)e(side)i(of)e(the)i(equations)f (are)g(en)o(tered)i(in)o(to)150 1885 y Fk(linsolv)o(e)p Fp(.)20 b(W)m(e)c(express)h(the)f(constan)o(ts)h(as)f(a)f(list)h(whic)o(h,)f(in)g (this)h(case,)h(has)f(only)150 1935 y(one)e(en)o(try)m(.)212 2002 y Fn(>)282 2006 y Fo(X)21 b(:=)h(linsolve\(A,[4]\);)212 2084 y Fp(The)13 b(arbitrary)f(constan)o(ts)h(that)f(Maple)g(has)g(in)o(tro)q (duced)h(are)p 1213 2084 13 2 v 28 w Fj(t)1241 2090 y Fi(1)1271 2084 y Fp(and)p 1353 2084 V 28 w Fj(t)1381 2090 y Fi(2)1399 2084 y Fp(.)18 b(The)150 2134 y(lead)13 b(c)o(haracter)i(is)e(the)g (underscore)j(\(not)d(a)g(min)o(us)f(sign\))h(and)g(the)g(n)o(um)o(b)q(ers)g (1)g(and)150 2184 y(2)h(are)g(subscripts.)19 b(T)m(o)14 b(address)h(these)g (constan)o(ts)g(in)e(Maple,)g(use)i(the)f(syn)o(tax)p 1428 2184 V 29 w(t[1])150 2234 y(and)p 233 2234 V 29 w(t[2].)212 2284 y(Setting)20 b(these)h(constan)o(ts)f(alternativ)o(ely)f(equal)g(to)g(0) g(and)g(1)g(will)f("extract")150 2334 y(the)e(basis)f(v)o(ectors)h(in)e(the)i (general)f(solution.)20 b(After)c(de\014ning)f Fk(u)p Fp(,)f(the)i (translation)150 2384 y(v)o(ector,)g(it)f(is)g(essen)o(tial)h(to)f(subtract)h Fk(u)f Fp(from)f Fk(X)h Fp(when)h(extracting)f(the)h(v)o(ectors)h Fk(v)150 2434 y Fp(and)d Fk(w)p Fp(,)g(the)g(m)o(ultipliers)e(of)p 619 2434 V 28 w Fj(t)646 2440 y Fi(1)679 2434 y Fp(and)p 762 2434 V 29 w Fj(t)790 2440 y Fi(2)808 2434 y Fp(.)p eop bop 450 275 a Fp(16)967 b Fm(1.3)13 b(Maple)h(On)g(Line)512 387 y Fn(>)582 391 y Fo(u)21 b(:=)h(subs\(_t[1]=0,)d(_t[2]=0,)h(op\(X\)\);)g (v)i(:=)f(subs\(_t[1]=1,)e(_t[2]=0,)512 434 y(evalm\(X-u\)\);)h(w)h(:=)h (subs\(_t[1]=0,)d(_t[2]=1,)h(evalm\(X-u\)\);)512 511 y Fp(T)m(o)10 b(get)g(Maple)g(to)g(write)g(the)h(general)f(solution)f(in)h(the)g(v)o(ector) h(form)d(giv)o(en)i(in)f(the)450 561 y(statemen)o(t)15 b(of)f(the)h(problem,) e(adroit)h(use)i(of)e(the)h Fk(ev)m(alm)f Fp(command)e(is)j(necessary)m(.)450 611 y(Unless)f Fk(ev)m(alm)f Fp(is)g(applied)f(to)h(the)h(v)o(ector)g Fk(v,)f Fp(the)h(screen)h(will)d(merely)g(displa)o(y)g(the)450 661 y(name,)g(v.)512 727 y Fn(>)582 731 y Fo(Xg)21 b(:=)h(evalm\(u\))e(+)h (s*evalm\(v\))f(+)i(t*evalm\(w\);)512 809 y Fp(a\))12 b(With)f(the)h(v)o (ectors)g Fk(v)g Fp(and)f Fk(w)h Fp(declared)h(as)f(ab)q(o)o(v)o(e,)f(\(or)g (t)o(yp)q(ed)i(in)e(afresh)h(from)450 859 y(the)i(p)q(encil-and-pap)q(er)h (solution\),)d(form)g Fk(C)p Fp(,)i(the)h(general)f(elemen)o(t)f(in)g(the)i (span)f(of)450 908 y Fk(v)g Fp(and)g Fk(w)p Fp(.)512 974 y Fn(>)582 978 y Fo(C)21 b(:=)h(evalm\(a*v)e(+)h(b*w\);)512 1056 y Fp(b\))14 b(Sho)o(w)g(that)g Fk(F)g Fp(=)g Fk(u)f Fp(+)h Fk(C)h Fp(solv)o(es)f(the)g(giv)o(en)g(equation)f Fj(x)c Fp(+)h(2)d Fj(y)j Fp(+)g(3)d Fj(z)13 b Fp(=)f(4.)512 1122 y Fn(>)582 1126 y Fo(F)21 b(:=)h(evalm\(u+C\);)512 1192 y Fn(>)582 1196 y Fo(q;)512 1262 y Fn(>)582 1266 y Fo(subs\(x=F[1],)d(y=F[2],)h(z=F[3],)h(q\);)512 1344 y Fp(This)16 b(sho)o(ws)f(that)h(the)g(general)g(solution)e(of)h(the)h (giv)o(en)f(equation)g(is)h(the)g(span)450 1394 y(of)d(the)i(v)o(ectors)g Fk(v)f Fp(and)g Fk(w)p Fp(,)f(translated)i(b)o(y)e(the)i(v)o(ector)g Fk(u)p Fp(.)512 1444 y(c\))21 b(Substitute)g(the)f(v)o(ector)h(C)e(in)o(to)h (the)g(giv)o(en)g(equation)f(and)h(describ)q(e)h(the)450 1494 y(result.)e(What)13 b(should)h(y)o(ou)f(conjecture)j(from)c(this)i(result?)19
b(Can)14 b(y)o(ou)f(pro)o(v)o(e)h(it?)450 1611 y Fe(Exercise)j(2)450 1689 y Fp(Rep)q(eat)c(Exercise)i(1)e(for)f(the)i(system)e(\(U\))i(from)d (Section)i(1.3.)k(The)c(essen)o(tial)g(ques-)450 1738 y(tion)i(to)g(b)q(e)h (ultimately)d(resolv)o(ed)k(is)e("Do)q(es)h(the)g(conjecture)h(made)d(in)h (Exercise)i(1)450 1788 y(still)c(hold?")g(Can)g(y)o(ou)h(then)g(pro)o(v)o(e)g (y)o(our)g(answ)o(er)g(is)g(correct?)512 1838 y(Begin)g(b)o(y)f(en)o(tering)h (the)h(system)e(\(U\),)g(b)q(eing)h(careful)g(to)f(clear)h(the)g(v)n(ariable) f Fk(w)450 1888 y Fp(that)h(w)o(as)g(used)h(in)e(Exercise)i(1,)512 1938 y(using)e(the)h(command)d(")i Fo(w:='w')p Fp(".)k(Call)12 b(y)o(our)h(equations)h(q1,)e(q2,)h(q3,)g(and)g(q4.)512 1988 y(Use)i Fk(genmatrix)d Fp(to)i(form)e(A,)h(the)i(co)q(e\016cien)o(t)f(matrix) e(for)i(the)g(system)g(\(U\).)512 2054 y Fn(>)582 2058 y Fo(A)21 b(:=)h(genmatrix\([q.\(1.)o(.4\)],)o([x,y,)o(z,w]\))o(;)512 2136 y Fp(Use)16 b Fk(linsolv)o(e)c Fp(to)j(obtain)g(the)g(general)h (solution)e(of)h(system)g(\(U\),)g(using)g(a)f(list)450 2186 y(for)g(the)g(v)n(alues)g(on)f(the)i(righ)o(t)e(hand)h(sides)h(of)e(the)h (equations.)512 2252 y Fn(>)582 2256 y Fo(X)21 b(:=)h(linsolve\(A,[1,0,)o (1,2]\))o(;)512 2334 y Fp(Extract)c(the)g(translation)e(v)o(ector)i Fk(u)e Fp(and)h(the)h(basis)f(v)o(ectors)h Fk(v)f Fp(and)g Fk(w)g Fp(as)g(in)450 2384 y(Exercise)f(1.)512 2434 y(a\))e(F)m(orm)e Fk(C)p Fp(,)i(the)g(general)h(elemen)o(t)e(in)g(the)i(span)f(of)f Fk(v)h Fp(and)g Fk(w)p Fp(.)p eop bop 150 275 a Fm(1.4)13 b(Maple)h(On)g(Line)967 b(17)212 387 y Fn(>)282 391 y Fo(C)21 b(:=)h(evalm\(a*v)e(+)h(b*w\);)212 472 y Fp(b\))c(Sho)o(w)g(that)g Fk(F)g Fp(=)g Fk(u)g Fp(+)g Fk(C)h Fp(satis\014es)g(the)f(system)g(\(U\).)g(This)g(can)g(b)q(e)h(done)150 522 y(b)o(y)13 b(rep)q(etitiv)o(e)h(t)o(yping,)e(b)o(y)g(t)o(yping)h(once)g (and)g(using)g(cop)o(y/paste,)g(or)g(b)o(y)g(a)g Fl(for-lo)n(op)150 572 y Fp(that)h(do)q(es)h(rep)q(etition)f(automatically)l(.)212 640 y Fn(>)282 644 y Fo(F)21 b(:=)h(evalm\(u)e(+)i(C\);)212 712 y Fn(>)282 716 y Fo(for)f(k)h(from)f(1)g(to)g(4)h(do)f(subs\(x=F[1],)f (y=F[2],)g(z=F[3],)g(w=F[4],)212 759 y(q.k\);)h(od;)212 839 y Fp(c\))16 b(Substitute)f(C)g(in)o(to)f(eac)o(h)h(equation)f(in)h(the)g (system)f(\(U\))h(in)g(an)f(attempt)g(to)150 889 y(determine)j(if)f(the)h (conjecture)h(made)e(in)g(Exercise)j(1)d(is)h(still)e(viable.)26 b(If)17 b(y)o(ou)f(still)150 939 y(b)q(eliev)o(e)e(y)o(our)g(conjecture)h(is) f(true,)h(can)f(y)o(ou)f(pro)o(v)o(e)h(it?)150 1082 y Fq(1.4)66 b(On)22 b(Line)150 1175 y Fp(After)15 b(restarting)g(Maple)g(and)f (reinitializing)f(b)o(y)h(loading)f(the)i Fl(linalg)f Fp(pac)o(k)n(age,)g(w)o (e)150 1224 y(examine)k(the)i Fk(rref)f Fp(command)d(for)j(putting)h(a)f (matrix)e(in)o(to)i(its)h(reduced)h(\(ro)o(w\))150 1274 y(ec)o(helon)14 b(form.)212 1325 y(Consider)h(the)f(matrix)e(A)i(de\014ned)h(b)o(y)212 1393 y Fn(>)282 1397 y Fo(A)21 b(:=)h(matrix\(4,6,)d([1,-1,1,3,0,6,2,-2,)o (2,6,0)o(,7,-)o(1,1,1)o(,-1,-)o(2,1,4)o(,-4,1)o(,9,3,)o(6]\);)212 1478 y Fp(Sub)r(ject)c(A)f(to)g(the)h Fk(rref)d Fp(op)q(erator.)212 1546 y Fn(>)282 1550 y Fo(rref\(A\);)212 1631 y Fp(The)k(result)h(just)f (obtained)f(is)h(exact)g(since)h(Maple)e(obtained)h(it)f(b)o(y)g(doing)g(ra-) 150 1681 y(tional)e(arithmetic.)k(There)f(is)e(no)g(truncation)g(error)h(in)o (tro)q(duced)g(b)o(y)f(a)g(con)o(v)o(ersion)150 1731 y(of)j(in)o(tegers)h(to) g(decimal)d(form,)i(and)g(there)i(is)e(no)g(round-o\013)h(error)g(pro)q (duced)h(b)o(y)150 1780 y(a)c(n)o(umerical)e(algorithm.)19 b(This)c(is)g(the)g(result)h(that)f(w)o(ould)f(b)q(e)i(obtained)f(w)o(orking) 150 1830 y(with)f(a)f(p)q(encil)h(and)g(pap)q(er.)212 1881 y(This)d(is,)g(of)f(course,)j(w)o(onderful.)j(W)m(e)11 b(can,)g(in)g (princple,)g(do)g(ev)o(ery)h(computation)150 1931 y(with)k(total)g(accuracy)m (.)26 b(A)o(t)17 b(\014rst)g(glance,)g(then,)g(it)f(seems)h(then)g(w)o(e)g (should)f(nev)o(er)150 1981 y(again)j(need)i(to)e(round)h(o\013)g(an)g(answ)o (er.)37 b(Unfortuately)m(,)20 b(life)f(is)h(not)g(so)g(simple.)150 2031 y(Supp)q(ose,)14 b(for)g(example,)e(w)o(e)i(w)o(an)o(t)g(MAPLE)g(to)g
(compute)f(1)p Fj(:)p Fp(048577)1268 2015 y Fi(20)1315 2031 y Fp(.)212 2081 y(W)m(e)h(en)o(ter)212 2150 y Fn(>)282 2154 y Fo(\(1048577/1000000)o(\)^20;)212 2234 y Fp(Maple's)d(resp)q(onse)i(is)e(a) g(121)f(digit)g(in)o(teger)h(divided)g(b)o(y)g(an)g(equally)f(large)h(p)q(o)o (w)o(er)150 2284 y(of)19 b(10.)33 b(Imagine)17 b(no)o(w,)j(what)f(w)o(ould)f (happ)q(en)i(if)e(w)o(e)i(attempted)f(to)g(p)q(erform)f(a)150 2334 y(calculation)9 b(whic)o(h)h(required,)h(sa)o(y)m(,)f(addition)f(and)h (m)o(ultiplicatio)o(n)e(sev)o(eral)i(h)o(undred)150 2384 y(suc)o(h)19 b(n)o(um)o(b)q(ers.)32 b(The)19 b(n)o(um)o(b)q(er)f(of)g(digits)g(our)g (computer)g(w)o(ould)g(need)i(to)e(store)150 2434 y(w)o(ould)f(b)q(ecome)g (astronomical)f(and)h(the)h(sp)q(eed)h(w)o(ould)e(b)q(e)h(reduced)h(to)f(a)f (snail's)p eop bop 450 275 a Fp(18)967 b Fm(1.4)13 b(Maple)h(On)g(Line)450 391 y Fp(pace.)26 b(F)m(urthermore,)17 b(in)f(an)g(actual)g(application,)g (the)h(n)o(um)o(b)q(er)f(.1048577)e(w)o(ould)450 441 y(probably)i(represen)o (t)i(the)f(result)g(of)f(some)g(measuremen)o(t)f(whic)o(h)h(itself)g(migh)o (t)e(b)q(e)450 491 y(accurate)f(only)e(to)g(within)g(the)h(giv)o(en)f(n)o(um) o(b)q(er)g(of)g(digits.)17 b(Th)o(us,)12 b(the)g(v)n(ast)f(ma)r(jorit)o(y)450 541 y(of)k(the)h(digits)f(that)g(our)h(compter)f(is)g(so)h(lab)q(oriously)e (computing)g(and)h(sa)o(ving)g(are)450 591 y(totally)10 b(meaningless.)15 b(The)d(moral)c(is)j(that)g(the)h(p)q(erils)f(of)f(n)o(umerical)f (computations)450 640 y(m)o(ust)k(b)q(e)h(faced.)450 777 y Fe(Exercise)j(1)450 860 y Fp(W)m(e)e(\014rst)h(examine)e(ho)o(w)g(to)h(con)o (v)o(ert)h(the)g(matrix)d(A)i(to)h(\015oating)e(p)q(oin)o(t)g(\(decimal\))450 910 y(form,)19 b(then)h(lo)q(ok)f(at)h(the)g(same)f(ro)o(w)g(reduction)i (done)f(n)o(umerically)e(instead)i(of)450 960 y(sym)o(b)q(olically)l(.)25 b(The)18 b(con)o(v)o(ersion)f(can)h(b)q(e)f(done)h(b)o(y)f Fl(c)n(onverting)g Fp(eac)o(h)g(elemen)o(t)g(to)450 1010 y(\015oating)c(p)q (oin)o(t)g(form)f(-)i Fk(map)f Fp(the)i(con)o(v)o(ert)f(op)q(erator)h(on)o (to)e(the)i(matrix)d(A)i(via)512 1086 y Fn(>)582 1090 y Fo(A1)21 b(:=)h(map\(convert,A,f)o(loat\))o(;)512 1182 y Fp(Ro)o(w)15 b(reduce)i(to)e(reduced)i(ec)o(helon)f(form)d(the)j(\015oating)f(p)q(oin)o(t) g(form)e(of)i(matrix)450 1231 y(A.)512 1308 y Fn(>)582 1312 y Fo(rref\(A1\);)512 1404 y Fp(F)m(or)10 b(the)i(matrix)c(A)j(there)h(is)e (no)h(di\013erence)h(in)e(the)h(reduced)i(ec)o(helon)e(form)d(when)450 1453 y(w)o(orking)h(n)o(umerically)m(.)15 b(This)10 b(will)f(not)h(alw)o(a)o (ys)f(b)q(e)i(the)g(case.)18 b(In)10 b(fact,)h(w)o(e)g(can)f(in)o(v)o(es-)450 1503 y(tigate)k(Maple's)f(n)o(umerics)h(b)o(y)g(a)f(stratagem)g(used)i(on)f (an)o(y)g(n)o(umerical)e(calculating)450 1553 y(device.)19 b(Compute)12 b(the)i(v)n(alue)e(of)h([)1020 1537 y Fi(1)p 1011 1544 34 2 v 1011 1567 a(99)1049 1553 y Fp(])7 b(99)g Fg(\000)i Fp(1,)j(and)h(successiv)o(ely)i(app)q(end)f(9's)f(b)q(oth)450 1603 y(inside)e(and)g(outside)g(the)h(brac)o(k)o(ets.)18 b(Ev)o(en)o(tually)m (,)10 b(there)i(will)e(b)q(e)h(enough)g(9's)g(so)g(that)450 1653 y(the)k(n)o(umeric)f(calculation)f(will)g(no)i(longer)f(yield)g(0.)19 b(That)c(giv)o(es)f(an)g(indication)g(of)450 1702 y(ho)o(w)g(accurate)i(the)f (computing)e(device)j(is.)k(T)m(o)14 b(force)h(Maple)g(to)f(ev)n(aluate)h (the)g(ex-)450 1752 y(pressions)i(in)e(\015oating)f(p)q(oin)o(t)h(form,)f (mak)o(e)g(one)i(of)f(the)h(n)o(um)o(b)q(ers)f(a)g(decimal.)21 b(F)m(or)450 1802 y(example,)12 b(use)j("1.0")d(rather)j(than)f(just)g("1")f (in)g(the)i(n)o(umerator)e(of)g(the)i(fraction.)512 1856 y(T)m(o)f(see)i(the) g(di\013erence)g(b)q(et)o(w)o(een)h(w)o(orking)c(n)o(umerically)g(and)i(sym)o (b)q(olically)c(in)450 1905 y(Maple,)j(c)o(hange)g(the)h(n)o(umerator)e(from) f("1.0")h(to)h(just)g(1.)19 b(Then,)14 b(Maple)g(will)f(ev)n(al-)450 1955 y(uate)j(the)f(expression)i(sym)o(b)q(olically)12 b(and)j(pro)q(duce)h (0.)22 b(The)15 b(round-o\013)h(error)g(only)450 2005 y(app)q(ears)f(when)f (w)o(orking)f(with)h(\015oating)e(p)q(oin)o(t)i(n)o(um)o(b)q(ers.)512 2059 y(It)g(is)g(p)q(ossible)f(to)h(v)n(ary)f(the)h(n)o(um)o(b)q(er)f(of)g (digits)g(with)h(whic)o(h)f(Maple)h(computes.)450 2108 y(This)g(is)g(done)g
(via)f(the)h(Digits)f(v)n(ariable)g(as)h(follo)o(ws.)512 2185 y Fn(>)582 2189 y Fo(Digits)20 b(:=)i(12;)512 2280 y Fp(T)m(est)16 b(Maple's)f(n)o(umeric)g(b)q(eha)o(vior)g(on)g(the)h(\015oating)f(p)q(oin)o (t)g(calculation)f(ab)q(o)o(v)o(e)450 2330 y(that)g(failed)f(to)h(yield)f(0.) 512 2384 y(Notice)22 b(that)g(with)f(more)g(digits)g(a)o(v)n(ailable,)g (Maple)h(escap)q(ed)h(the)f(e\013ect)i(of)450 2434 y(round-o\013)18 b(in)g(a)g(computation)f(that)h("failed")f(with)h(just)g(the)h(default)f(10)g (digits.)p eop bop 150 275 a Fm(1.4)13 b(Maple)h(On)g(Line)967 b(19)150 391 y Fp(The)20 b(price)g(one)f(pa)o(ys)h(for)e(increasing)i(the)g(n)o(um)o(b)q (er)e(of)h(w)o(orking)g(digits)f(is)h(com-)150 441 y(putation)e(time,)f (since)i(these)g(extra)g(digits)e(are)i(b)q(eing)f(sim)o(ulated)e(b)o(y)i (the)h(Maple)150 491 y(soft)o(w)o(are.)212 543 y(Reset)d(the)g(n)o(um)o(b)q (er)e(of)g(digits)g(bac)o(k)h(to)g(the)g(default)g(10)f(via)212 614 y Fn(>)282 618 y Fo(Digits)20 b(:=)i(10;)150 746 y Fe(Exercise)17 b(2)150 826 y Fp(Use)g(the)g Fk(rref)e Fp(command)e(to)j(\014nd)h(all)e (solutions)g(to)i(the)f(system)g(in)g(Exercise)i(5g,)150 876 y(Section)c(1.3.)212 928 y(First,)j(en)o(ter)g(the)g(equations)f(of)g(that)g (system.)25 b(Call)15 b(y)o(our)h(equations)g(q1,)g(q2,)150 978 y(q3,)d(qnd)h(q4.)212 1030 y(Next,)i(get)f(Maple)g(to)g(write)h(the)f (augmen)o(ted)g(system)f(matrix.)20 b(The)c Fk(genma-)150 1080 y(trix)11 b Fp(command)f(con)o(v)o(erts)k(the)f(equations)f(in)o(to)g(matrix) f(form,)f(and)j(the)g(additional)150 1129 y(parameter)j Fl(\015ag)i Fp(signals)e(Maple)g(to)h(include)g(the)g(n)o(um)o(b)q(ers)f(on)h(the)g(righ) o(t)f(side)i(of)150 1179 y(the)g(equations.)27 b(Inciden)o(tally)m(,)17 b(the)g(parameter)g(can)g(b)q(e)h(an)o(y)f(c)o(haracter)h(or)f(w)o(ord)150 1229 y(that)d(is)g(not)f(already)h(a)g(reserv)o(ed)i(w)o(ord)d(in)h(Maple.) 212 1301 y Fn(>)282 1305 y Fo(A:=genmatrix\([q.)o(\(1..4)o(\)],[x)o(,y,z,)o (w],fl)o(ag\);)212 1390 y Fp(Apply)g(the)g Fk(rref)f Fp(command.)212 1461 y Fn(>)282 1465 y Fo(A1:=rref\(A\);)212 1550 y Fp(T)m(o)d(obtain)f (solutions)h(from)e(the)j(rref)g(form)d(of)i(the)h(matrix)d(A,)i(apply)f(the) i(pro)q(cess)150 1600 y(of)k(bac)o(k)g(substitution.)22 b(Start)15 b(with)g(the)h(b)q(ottom-most)d(non-zero)i(ro)o(w)g(of)g(rref\(A\))150 1650 y(and)e(in)o(terpret)i(it)e(as)g(an)g(equation)g(de\014ning)h(the)g(v)n (alue)e(of)h Fl(z)p Fp(.)18 b(Solv)o(e)13 b(that)g(equation)150 1700 y(for)h(the)h(v)n(alue)e(of)h Fl(z)g Fp(and)g(substitute)h(that)f(v)n (alue)g(in)o(to)f(the)i(equation)f(ab)q(o)o(v)o(e.)k(Solv)o(e)150 1750 y(the)12 b(resulting)g(equation)f(for)g(the)i(v)n(alue)d(of)h Fl(y)h Fp(so)g(determined.)17 b(Substitute)c(b)q(oth)e(the)150 1799 y(v)n(alue)i(of)h Fl(y)g Fp(and)f Fl(z)h Fp(in)o(to)f(the)i(remaining)d (equation)h(whic)o(h)h(is)g(then)g(solv)o(ed)g(for)f Fl(x)p Fp(.)212 1851 y(Chec)o(k)i(y)o(our)e(w)o(ork)h(b)o(y)g(in)o(v)o(oking)e (Maple's)h(built-in)g Fk(bac)o(ksub)f Fp(command.)212 1923 y Fn(>)282 1927 y Fo(backsub\(A1\);)150 2054 y Fe(Exercise)17 b(3)150 2135 y Fp(The)i(rank)g(of)f(a)g(system)g(of)g(equations)h(is)g(the)g (n)o(um)o(b)q(er)f(of)g(equations)g(left)h(after)150 2185 y(eliminating)12 b(dep)q(enden)o(t)k(equations.)21 b(This)15 b(n)o(um)o(b)q(er)f(do)q(es)i (not)e(dep)q(end)i(on)f(whic)o(h)150 2234 y(equations)k(w)o(ere)h(k)o(ept)g (or)f(eliminated.)32 b(Hence,)22 b(it)d(is)g(plausible)g(that)g(the)h(ro)o (ws)150 2284 y(in)13 b(the)i(reduced)g(ro)o(w)f(ec)o(helon)g(form)e(of)i(the) g(system's)g(matrix)e(re\015ect)j(the)g(distinct)150 2334 y(equations)10 b(that)g(w)o(ould)e(surviv)o(e)j(an)e(elimination)e(of)i(dep)q(enden)o(t)i (equations.)17 b(Hence,)150 2384 y(the)10 b(rank)g(of)f(the)i(system)e (should)h(b)q(e)g(the)h(n)o(um)o(b)q(er)e(of)g(non-zero)h(ro)o(ws)g(in)g(the) g(reduced)150 2434 y(ro)o(w)k(ec)o(helon)g(form)e(of)i(the)g(matrix)e(for)i (the)g(system.)p eop bop 450 275 a Fp(20)967 b Fm(1.4)13 b(Maple)h(On)g(Line)512 391 y Fp(Chec)o(k)j(this)e(conjecture)j(exp)q(erimen)o(tally)c(b)o(y)h (creating)h(\(4)g(x)f(5\))g(matrices)h(A1,)450 441 y(A2,)11
b(A3,)h(and)f(A4)h(with)f(ranks)h(resp)q(ectiv)o(ely)h(1,)f(2,)f(3,)g(4.)17 b(In)12 b(particular,)g(insure)g(that)450 491 y(no)i(matrix)e(has)i(a)f(zero) i(en)o(try)m(.)512 541 y(A)g(pro)q(cess)h(for)f(creating)g(a)f(random)f (matrix)g(of)h(prescrib)q(ed)j(rank)d(is)h(based)g(on)450 591 y(forming)c(ro)o(ws)i(that)g(are)h(themselv)o(es)f(linear)g(com)o(binations)e (of)i(other)h(ro)o(ws.)k(Begin)450 641 y(b)o(y)d(de\014ning)g(f,)g(a)f (function)h(returning)h(a)f(random)e(in)o(teger)j(in)e(the)i(closed)g(in)o (terv)n(al)450 691 y([-10,10].)f(This)f(is)g(done)g(with)g(the)g Fk(rand)f Fp(command.)512 757 y Fn(>)582 761 y Fo(f)21 b(:=)h (rand\(-10..10\):)512 840 y Fp(W)m(e)11 b(b)q(egin)h(b)o(y)f(constructing)h (a)f(matrix)f(A1,)h(of)g(rank)g(1.)17 b(This)11 b(requires)i(that)e(the)450 890 y(ro)o(ws)e(of)g(A1)g(b)q(e)h(linear)f(com)o(binations)e(of)i(a)g(single) g(ro)o(w.)16 b(Begin)10 b(b)o(y)f(constructing)h(a)f(\(1)450 940 y(x)h(5\))g(matrix)f(M1)h(b)o(y)g(using)g(the)h(function)f(f)g(in)g (conjunction)g(with)g(the)h Fk(randmatrix)450 989 y Fp(command)d(to)i(guaran) o(tee)i(that)e(the)i(random)d(matrix)g(has)i(en)o(tries)g(that)g(are)g(in)o (tegers)450 1039 y(in)i(the)i(in)o(terv)n(al)e([-10,10].)512 1106 y Fn(>)582 1110 y Fo(M1)21 b(:=)h(randmatrix\(1,5,)o(entri)o(es=f\))o(;) 512 1189 y Fp(F)m(rom)15 b(M1,)i(build)f(a)g(\(2)h(x)f(5\))h(matrix)d(M2)j (in)f(whic)o(h)h(the)g(ro)o(ws)g(are)g(m)o(ultiples)450 1238 y(of)h(the)h(single)f(ro)o(w)g(in)g(M1.)32 b(Maple's)18 b Fk(stac)o(k)g Fp(command)e(assem)o(bles)i(a)g(ro)o(ws)h(\(or)450 1288 y(v)o(ectors\))13 b(in)o(to)f(a)g(new)h(matrix,)d(making)g(the)i(building-blo)q(c)o(ks)f(in)o (to)h(the)g(ro)o(ws)h(of)e(the)450 1338 y(new)j(matrix.)512 1405 y Fn(>)582 1409 y Fo(M2)21 b(:=)h(stack\(M1,evalm\()o(f\(\)*r)o(ow\(M1)o (,1\)\)\))o(;)512 1487 y Fp(F)m(rom)9 b(M2,)i(build)e(a)i(\(3)f(x)g(5\))h (matrix)e(M3)h(in)g(whic)o(h)h(the)g(ro)o(ws)f(are)h(random)e(linear)450 1537 y(com)o(binations)j(of)i(the)h(ro)o(ws)g(in)f(M2.)20 b(A)14 b(single)h(ro)o(w)f(in)g(M2)g(can)h(b)q(e)g(referenced)i(b)o(y)450 1587 y(the)d Fk(ro)o(w)g Fp(command)d(as)j(illustrated)g(b)q(elo)o(w.)512 1653 y Fn(>)582 1657 y Fo(M3)21 b(:=)h(stack\(M2,evalm\()o(f\(\)*r)o(ow\(M2)o (,1\)+f)o(\(\)*ro)o(w\(M2)o(,2\)\)\))o(;)512 1736 y Fp(Finally)m(,)11 b(build)i(the)h(required)g(\(4)f(x)h(5\))f(matrix)e(A1)j(b)o(y)f(taking)f (linear)h(com)o(bina-)450 1786 y(tions)h(of)f(the)h(ro)o(ws)h(of)e(M3.)512 1853 y Fn(>)582 1857 y Fo(A1)21 b(:=)h(stack\(M3,evalm\()o(f\(\)*r)o(ow\(M3)o (,1\))d(+)i(f\(\)*row\(M3,2\))e(+)j(f\(\)*row\(M3,3\)\)\);)512 1935 y Fp(T)m(est)15 b(that)f(A1)f(has)h(rank)g(1)g(b)o(y)f(in)o(v)o(oking)f (Maple's)i(built-in)f Fk(rank)g Fp(command.)512 2002 y Fn(>)582 2006 y Fo(rank\(A1\);)512 2084 y Fp(The)h(pro)q(cess)h(for)e(constructing)h (random)e(matrices)h(of)g(rank)g(2)g(is)g(similar.)j(The)450 2134 y(only)g(di\013erence)k(is)d(that)g(w)o(e)g(start)h(with)f(a)g(random)f (\(2)h(x)g(5\))g(matrix)f(\(pro)q(duced)450 2184 y(using)g(the)h Fk(randmatrix)d Fp(command\))f(instead)k(of)f(a)g(\(1)g(x)g(5\))g(matrix.)23 b(Similarly)l(,)450 2234 y(for)13 b(a)h(rank)f(3)h(matrix,)d(w)o(e)j(w)o (ould)f(b)q(egin)h(with)f(a)g(random)f(\(3)i(x)f(5\))h(matrix)e(and)h(for)450 2284 y(a)h(rank)f(4)h(matrix)e(w)o(e)i(w)o(ould)f(b)q(egin)h(with)g(a)f (random)g(\(4)g(x)h(5\))g(matrix.)512 2334 y(The)j(rank)f(of)g(the)h (matrices)f(A1,)g(A2,)g(A3)g(and)g(A4)g(can)h(b)q(e)g(corrob)q(orated)g(b)o (y)450 2384 y(reducing)d(eac)o(h)f(to)g(reduced)i(ec)o(helon)f(form.)i(The)d (follo)o(wing)d(lo)q(op)j(implem)o(en)o(ts)e(the)450 2434 y(required)k (calculations.)p eop bop 150 275 a Fm(1.4)13 b(Maple)h(On)g(Line)967 b(21)212 387 y Fn(>)282 391 y Fo(for)21 b(k)h(from)f(1)g(to)g(4)h(do)f(rref\(A.k\);)f(od;) 212 469 y Fp(In)15 b(eac)o(h)g(case)h(the)f(n)o(um)o(b)q(er)f(of)g(distinct)h (non-zero)g(ro)o(ws)g(exactly)g(matc)o(hes)f(the)150 519 y(kno)o(wn)e(rank)h (of)f(the)h(matrix.)j(These)e(ro)o(w)e(reductions)i(are)f(exact,)g(without)f (round-)150 569 y(o\013)17 b(error)h(since)g(Maple)f(computes)g(sym)o(b)q (olically)d(unless)k(told)f(otherwise.)28 b(Ho)o(w-)150 619 y(ev)o(er,)15 b(all)e(computing)f(devices,)j(when)g(computing)d(with)i
(\015oating)f(p)q(oin)o(t)h(n)o(um)o(b)q(ers,)150 669 y(can)i(exp)q(erience)h (di\016culties)e(attributable)h(to)f(round-o\013)g(and)h(truncation)f (errors.)150 718 y(Examine)d(this)i(issue)h(in)e(Maple.)212 785 y Fn(>)282 789 y Fo(for)21 b(k)h(from)f(1)g(to)g(4)h(do)f (rref\(map\(convert,A.k)o(,flo)o(at\)\);)d(od;)212 867 y Fp(F)m(or)13 b(the)h(matrices)f(created)i(in)e(this)h(session)g(\(remem)o(b)q(er,)e(w)o(e) i(are)g(using)f(a)g(ran-)150 916 y(dom)f(pro)q(cess\),)k(rref\(A3\))e(is)g (wrong.)k(The)d Fk(rref)d Fp(command)f(declares)16 b(that)e(a)f(small)150 966 y(n)o(um)o(b)q(er)g(whic)o(h)h(ough)o(t)f(to)h(b)q(e)h(seen)g(as)f(zero,) g(is)g(not)g(zero.)19 b(Hence,)c(it)e(suggests)i(the)150 1016 y(rank)f(of)f(A3)h(is)g(four.)212 1066 y(One)19 b(defense)h(against)e(suc)o (h)h(n)o(umeric)f(errors)i(is)e(the)h Fk(gausselim)d Fp(command)150 1116 y(whic)o(h)h(ro)o(w)h(reduces)h(a)e(matrix)f(but)h(do)q(es)i(not)e(mak)o (e)f(the)i(diagonal)d(elemen)o(ts)j(1.)150 1166 y(By)f(not)g(dividing)f(b)o (y)h(the)h(diagonal)d(elemen)o(ts,)i(this)g(command)e(is)i(less)g(lik)o(ely)f (to)150 1216 y(err)h(in)e(n)o(umeric)g(computations.)23 b(Let)16 b(B3)g(b)q(e)h(the)f(\015oating)f(p)q(oin)o(t)g(v)o(ersion)h(of)g(A3,)150 1265 y(obtained)e(b)o(y)f(mapping)f(the)i(pro)q(cess)i(of)d(con)o(v)o(ersion) i(to)e(\015oats)h(on)o(to)g(A.)212 1332 y Fn(>)282 1336 y Fo(B3)21 b(:=)h(map\(convert,A3,)o(float)o(\);)212 1413 y Fp(No)o(w)14 b(apply)f Fk(gausselim)p Fp(.)212 1480 y Fn(>)282 1484 y Fo(q)21 b(:=)h(gausselim\(B3\);)212 1562 y Fp(The)17 b(small)c(en)o(tries)k(in)e(the) i(fourth)e(ro)o(w)h(should)g(b)q(e)g(tak)o(en)g(as)g(0's.)23 b(These)18 b(are)150 1611 y(the)h(n)o(um)o(b)q(ers)f(that)g(rref)h(sees)h(as) f(non-zero,)g(leading)f(to)g(errors.)33 b(In)18 b(Maple,)h(w)o(e)150 1661 y(can)f(apply)e(the)i Fk(fnormal)d Fp(command)g(to)i(set)i(to)e(zero)h (n)o(um)o(b)q(ers)f(smaller)f(than)h(a)150 1711 y(giv)o(en)11 b(tolerance.)18 b(As)13 b(with)e(all)g(op)q(erations)h(applied)f(to)h (matrices)f(and)g(v)o(ectors,)i(the)150 1761 y Fk(fnormal)f Fp(command)f(is)j(mapp)q(ed)f(on)o(to)g(the)h(matrix)e(q.)212 1827 y Fn(>)282 1831 y Fo(map\(fnormal,q,10)o(\);)150 1949 y Fe(Exercise)17 b(4)150 2026 y Fp(Ro)o(w)11 b(reduce)i(the)g(transp)q(oses)g (of)f(the)g(matrices)g(A1,)f(A2,)h(A3,)f(and)h(A4)g(constructed)150 2076 y(in)j(Exercise)i(3.)23 b(The)16 b(Maple)f(command)e(for)i(the)h(transp) q(ose)h(is)f(")f(transp)q(ose\(A\);")150 2126 y(What)e(do)h(y)o(ou)g(notice)g (ab)q(out)g(the)g(rank)g(of)f(the)i(resulting)f(matrices?)150 2244 y Fe(Exercise)j(5)150 2321 y Fp(De\014ne)e(v)o(ectors)g Fk(X)p Fp(,)e Fk(Y)p Fp(,)h(and)f Fk(Z)h Fp(as)g(indicated)g(b)q(elo)o(w.)212 2387 y Fn(>)282 2391 y Fo(X)21 b(:=)h(vector\([1,2,-5,4)o(,3]\);)c(Y)k(:=)f (vector\([6,1,-8,2,1)o(0]\);)d(Z)212 2434 y(:=)k(vector\([-5,12,-1)o(9,24,)o (1]\);)p eop bop 450 275 a Fp(22)967 b Fm(1.5)13 b(Maple)h(On)g(Line)512 391 y Fp(a\))19 b(Determine)f(whic)o(h)g(of)g(the)h(v)o(ectors)h Fk(U)e Fp(and)g Fk(V)h Fp(b)q(elo)o(w)f(is)h(in)f(the)h(span)g(of)450 441 y Fk(X)p Fp(,)d Fk(Y)p Fp(,)g(and)g Fk(Z)p Fp(.)h(Solv)o(e)f(a)g (determining)f(system)h(of)g(equations)h(b)o(y)f(using)g(the)h Fk(rref)450 491 y Fp(command.)512 562 y Fn(>)582 566 y Fo(U)k(:=)h (vector\([-5,23,-4)o(1,46,)o(8]\);)c(V)k(:=)f(vector\([22,0,-22,0,)o(34]\);) 512 649 y Fp(The)15 b(question)g(requires)g(solving)f(a)g Fk(X)g Fp(+)h(b)f Fk(Y)h Fp(+)f(c)h Fk(Z)g Fp(=)f Fk(U)p Fp(,)g(and)g(a)g Fk(X)h Fp(+)f(b)h Fk(Y)450 699 y Fp(+)f(c)g Fk(Z)g Fp(=)h Fk(V)f Fp(for)f(constan)o(ts)i(a,)e(b,)h(and)f(c.)19 b(Both)14 b(sets)h(of)f (equations)f(can)i(b)q(e)f(solv)o(ed)450 749 y(at)g(the)g(same)f(time)g(if)g (the)h(follo)o(wing)e(augmen)o(ted)h(matrix)f(is)h(formed.)512 820 y Fn(>)582 824 y Fo(q)21 b(:=)h(augment\(X,Y,Z,U,)o(V\);)512 908 y Fp(Ro)o(w)14 b(reducing)h(via)f(the)h(rref)g(command)d(giv)o(es)i (solutions)h(to)f(b)q(oth)h(systems)g(of)450 957 y(equations)f(at)g(the)g (same)f(time.)512 1009 y(b\))h(Imagine)d(that)i(y)o(ou)g(are)h(the)f(head)h (of)f(an)g(engineering)g(group)g(and)g(that)g(y)o(ou)450 1059
y(ha)o(v)o(e)g(a)f(computer)h(tec)o(hnician)g(w)o(orking)f(for)h(y)o(ou)g (who)f(kno)o(ws)h(absolutely)g(nothing)450 1109 y(ab)q(out)g(linear)f (algebra,)g(other)i(than)e(ho)o(w)h(to)f(en)o(ter)i(matrices)f(and)f (commands)f(in)o(to)450 1159 y(Maple.)17 b(Y)m(ou)12 b(need)g(to)g(tell)g(y)o (our)f(tec)o(hnician)i(ho)o(w)e(to)h(do)g(problems)f(similar)e(to)j(part)450 1208 y(\(a\))j(ab)q(o)o(v)o(e.)20 b(Sp)q(eci\014cally)15 b(,)g(y)o(ou)f(will) f(giv)o(e)i(the)g(tec)o(hnician)g(an)g(initial)e(set)i(of)g(three)450 1258 y(v)o(ectors,)g Fk(X)p Fp(,)e Fk(Y)p Fp(,)h(and)g Fk(Z)g Fp(from)e Fj(R)980 1243 y Fi(3)999 1258 y Fp(.)18 b(Y)m(ou)13 b(will)g(then)h(pro)o(vide)g(an)g(additional)e(v)o(ector)450 1308 y Fk(U)j Fp(and)h(y)o(ou)f(w)o(an)o(t)g(the)h(tec)o(hnician)g(to)f (determine)h(whether)h Fk(U)e Fp(is)g(in)h(the)g(span)g(of)450 1358 y Fk(X)p Fp(,)d Fk(Y)p Fp(,)h(and)g Fk(Z)p Fp(.)512 1409 y(W)m(rite)d(a)f(brief)h(set)h(of)e(instructions)i(whic)o(h)f(will)e(tell)i (y)o(our)g(tec)o(hnician)g(ho)o(w)f(to)h(do)450 1459 y(this)k(job.)k(Be)d(as) f(explicit)f(as)g(p)q(ossible.)21 b(Remem)o(b)q(er)13 b(that)i(the)g(tec)o (hnician)g(cannot)450 1509 y(do)g(linear)g(algebra!)21 b(Y)m(ou)15 b(m)o(ust)f(pro)o(vide)h(instructions)h(on)f(ho)o(w)g(to)g(construct)i(the) 450 1559 y(necessary)c(matrices,)d(what)h(to)f(do)h(with)f(them)g(and)g(ho)o (w)h(to)f(in)o(terpret)i(the)f(answ)o(ers.)450 1609 y(The)f(\014nal)g ("output")f(to)h(y)o(ou)g(should)f(b)q(e)i(a)e(simple)g("Y)m(es")h(or)g ("No.")f(Y)m(ou)g(don't)g(w)o(an)o(t)450 1659 y(to)14 b(see)h(matrices.)512 1710 y(c\))f(One)g(of)e(y)o(our)h(assistan)o(t)g(engineers)i(commen)o(ts)c (that)i(it)g(w)o(ould)f(b)q(e)h(easier)h(for)450 1760 y(the)g(tec)o(hnician)f (in)g(part)g(\(b\))h(to)f(use)h(Maple's)e Fk(rank)h Fp(command)e(rather)j (than)f Fk(rref)p Fp(.)450 1810 y(What)g(do)q(es)i(y)o(our)f(assistan)o(t)g (ha)o(v)o(e)g(in)f(mind?)450 1957 y Fq(1.5)66 b(On)22 b(Line)450 2051 y Fp(After)e(clearing)g(Maple's)f(memory)f(b)o(y)h(issuing)g(a)h Fk(restart)e Fp(command,)g(and)h(re-)450 2101 y(initializing)9 b(b)o(y)i(loading)f(the)j Fl(linalg)e Fp(pac)o(k)n(age,)g(en)o(ter)i(the)f (matrix)e(A)i(and)f(the)h(v)o(ector)450 2151 y Fk(X)p Fp(.)512 2221 y Fn(>)582 2225 y Fo(A)21 b(:=)h(matrix\(3,4,[1,2,)o(1,3,-)o(5,7,2)o (,2,13)o(,4,4,)o(3]\);)512 2296 y Fn(>)582 2300 y Fo(X)f(:=)h (vector\([1,3,-2,4)o(]\);)512 2384 y Fp(Obtain)13 b(the)g(pro)q(duct)g Fk(B)g Fp(=)f(A)h Fk(X)p Fp(.)f(\(Y)m(ou)g(should)h(consult)g(On)g(Line)f (Section)h(1.1)450 2434 y(for)h(a)f(discussion)i(of)e(matrix)f(pro)q(ducts)j (in)e(Maple.\))p eop bop 150 275 a Fm(1.5)13 b(Maple)h(On)g(Line)967 b(23)212 387 y Fn(>)282 391 y Fo(B:=evalm\(A&*X\);)150 540 y Fe(Exercise)17 b(1)150 618 y Fp(The)i(matrix)e(m)o(ultiplication)e(A)k Fk(X)g Fp(just)g(obtained)g(represen)o(ts)i(a)e(linear)f(com)o(bi-)150 667 y(nation)i(of)f(the)i(columns)e(of)h(A,)g(with)g(co)q(e\016cien)o(ts)h (tak)o(en)g(from)d(the)j(v)o(ector)g Fk(X)p Fp(.)150 717 y(Implemen)o(t)15 b(this)i(notion,)f(and)h(sho)o(w)g(the)h(result)g(is)f(the)g(v)o(ector)h Fk(B)f Fp(found)g(in)f(the)150 767 y(In)o(tro)q(duction.)212 817 y(Columns)e(of)h(A)g(can)h(b)q(e)g(referenced)i(with)d(the)h Fk(col)e Fp(command,)f(and)i(elemen)o(ts)150 867 y(of)j(the)i(v)o(ector)f Fk(X)g Fp(can)h(b)q(e)f(referenced)j(as)d Fk(X)p Fp([k].)32 b(Hence,)21 b(the)e(brute)h(force)g(w)o(a)o(y)150 917 y(of)c(obtaining)f(the) i(required)h(linear)e(com)o(bination)e(w)o(ould)i(b)q(e)h(with)f(the)i(follo) o(wing)150 967 y(syn)o(tax.)212 1034 y Fn(>)282 1038 y Fo (evalm\(col\(A,1\)*X)o([1])h(+)i(col\(A,2\)*X[2])e(+)j(col\(A,3\)*X[3])d(+)j (col\(A,4\)*X[4]\);)212 1117 y Fp(Since)c(the)g(columns)e(of)g(A)h(are)h (referenced)i(in)c(n)o(umerical)g(order)i(with)f(an)g(in-)150 1167 y(dex)g(that)g(is)f(rep)q(eated)i(when)f(referencing)h(the)f(comp)q (onen)o(ts)g(of)f(X,)g(it)g(should)g(b)q(e)150 1217 y(p)q(ossible)e(to)f (form)e(the)j(same)f(linear)g(com)o(bination)d(of)j(columns)f(with)h(some)g (sort)h(of)150 1267 y(summation)8 b(pro)q(cess.)19 b(Maple)11 b(has)h(a)f Fk(sum)f Fp(command)f(that)j(replicates)g(exactly)f(the)150
1316 y(mathematical)e(sigma)i(notation,)702 1285 y Ff(P)746 1296 y Fi(4)746 1329 y Fn(k)q Fi(=1)822 1316 y Fj(A)853 1322 y Fn(k)881 1316 y Fj(X)915 1322 y Fn(k)936 1316 y Fp(.)17 b(There)d(is)e(one) h(syn)o(tactical)f(quirk)150 1366 y(to)18 b(o)o(v)o(ercome,)g(ho)o(w)o(ev)o (er.)30 b(The)18 b Fk(col)g Fp(command)d(requires)k(a)f(v)n(alue)f(for)g(the) i(index)150 1416 y(b)q(efore)c(the)g Fk(sum)f Fp(command)e(can)j(pro)o(vide)f (it,)g(so)g(naiv)o(e)g(use)i(of)e(the)h(notation)e(will)150 1466 y(result)k(in)f(a)f(syn)o(tax)h(error.)26 b(The)17 b(tric)o(k)f(is)g(to) g(put)g(single)g(forw)o(ard)g(quotes)g(on)g(the)150 1516 y Fk(col)e Fp(command,)e(thereb)o(y)k(prev)o(en)o(ting)g(it)e(from)f(demanding) h(priorit)o(y)g(in)g(getting)h(a)150 1566 y(v)n(alue)e(of)h(the)g(index)g(b)q (efore)h(the)f Fk(sum)f Fp(command)e(is)j(ready)g(to)g(pro)o(vide)g(it.)212 1633 y Fn(>)282 1637 y Fo(evalm\(sum\('col\(A)o(,k\)'*)o(X[k],)o(k=1..)o (4\)\);)150 1785 y Fe(Exercise)j(2)150 1863 y Fp(Solv)o(e)g(the)h(system)f(A) h Fk(X)f Fp(=)h Fk(B)f Fp(for)g Fk(X)p Fp(.)g(Keep)i(in)e(mind)f(that)h Fk(B)h Fp(w)o(as)f(formed)f(b)o(y)150 1913 y(m)o(ultiplying)11 b(A)k(against)f Fk(X)p Fp(.)h(This)g(exercise)h(seeks)h(to)d(determine)h (whether)h(or)f(not)150 1963 y Fk(X)f Fp(can)g(b)q(e)h(reco)o(v)o(ered)g (from)d Fk(B)p Fp(.)212 2013 y(One)i(metho)q(d)e(of)h(solution)f(consists)j (of)d(ro)o(w)h(reducing)h(the)g(augmen)o(ted)e(matrix)150 2063 y([A,)p Fk(B)p Fp(],)c(then)i(using)f(bac)o(k)g(substitution,)h(implemen)o (ted)d(in)i(Maple)h(via)e(the)i Fk(bac)o(ksub)150 2113 y Fp(command.)212 2180 y Fn(>)282 2184 y Fo(C)21 b(:=)h(rref\(augment\(A,B)o(\)\);)212 2251 y Fn(>)282 2255 y Fo(X1)f(:=)h(backsub\(C\);)212 2334 y Fp(By)17 b(insp)q(ection,)f(determine)h(a)e(v)n(alue)h(of)f(the)i (parameter)p 1161 2334 13 2 v 31 w Fj(t)1188 2340 y Fi(1)1223 2334 y Fp(that)f(mak)o(es)f(the)150 2384 y(general)f(solution)e(in)h(X1)g(b)q (ecome)g(precisely)h Fk(X)p Fp(.)f(Remem)o(b)q(er,)f(this)h(parameter)g(is)g (a)150 2434 y(subscripted)j(quan)o(tit)o(y)m(,)c(and)h(can)i(b)q(e)f (addressed)i(in)d(Maple)h(via)f(the)h(syn)o(tax)p 1396 2434 V 29 w(t[1].)p eop bop 450 275 a Fp(24)967 b Fm(1.5)13 b(Maple)h(On)g(Line)450 391 y Fe(Exercise)j(3)450 469 y Fp(Another)f(metho)q(d)d(for)i(\014nding)f (the)h(general)g(solution)f(\014rst)i(obtained)e(in)g(Exercise)450 518 y(3)f(is)g(predicated)i(on)e(\014nding)g(a)g(basis)g(for)g(the)h(n)o(ull) e(space)j(of)d(A.)h(This)g(basis)h(can)f(b)q(e)450 568 y(found)g(via)g(the)i (Maple)f(command)d Fk(n)o(ullsp)o(ace)p Fp(,)g(as)j(sho)o(wn)g(b)q(elo)o(w.) 512 634 y Fn(>)582 638 y Fo(q)21 b(:=)h(nullspace\(A\);)512 716 y Fp(Observ)o(e)c(that)f(the)g Fk(n)o(ullsp)o(ace)d Fp(command)g(returns) k(a)e(set)h(of)f(v)o(ectors.)27 b(Here,)450 766 y(there)17 b(is)e(but)h(one)f(mem)o(b)q(er)f(in)h(the)h(set,)g(a)f(single)g(v)o(ector)h (that)g(can)f(b)q(e)h(addressed)450 816 y(via)d(the)i(syn)o(tax)512 882 y Fn(>)582 886 y Fo(Z:=q[1];)512 964 y Fp(V)m(eri\014cation)c(that)g Fk(Z)g Fp(is)g(indeed)g(in)g(the)g(n)o(ull)f(space)i(of)f(A)g(resides)h(in)e (the)i(pro)q(duct)450 1013 y(A)i Fk(Z)p Fp(.)512 1079 y Fn(>)582 1083 y Fo(evalm\(A)20 b(&*)i(Z\);)512 1161 y Fp(The)16 b(general)f(solution)f (for)g(the)i(system)f(A)g Fk(X)g Fp(=)g Fk(B)g Fp(is)g(then)g Fk(Xg)g Fp(=)g Fk(X)g Fp(+)g(t)g Fk(Z)p Fp(,)450 1211 y(where)g(t)f(is)g(an)g (arbitrary)f(parameter.)18 b(F)m(orm)12 b Fk(Xg)i Fp(and)g(sho)o(w)g(that)g (it)f(satis\014es)i(the)450 1261 y(equation)f(A)f Fk(Xg)h Fp(=)g Fk(B)p Fp(.)512 1311 y(By)f(insp)q(ection,)g(determine)f(a)h(v)n(alue)e(of)h (the)h(parameter)g(t)f(in)g(Xg)h(for)f(whic)o(h)g(Xg)450 1361 y(b)q(ecomes)i(exactly)g(X1,)f(the)i(\014rst)g(form)d(of)h(the)h(general)h (solution)e(found)g(ab)q(o)o(v)o(e.)450 1479 y Fe(Exercise)k(4)450 1556 y Fp(En)o(ter)e(the)f(matrix)e(A)i(and)g(the)h(v)o(ector)f Fk(B)g Fp(as)g(sho)o(wn)g(b)q(elo)o(w.)512 1622 y Fn(>)582 1626 y Fo(A)21 b(:=)h(matrix\(4,6,[17,-)o(6,13,)o(27,64)o(,19,4)o(,-6,-)o (33,2)o(5,7,9)o(,55,-)o(24,6,)o(106,1)o(99,66)o(,89,)o(-)512 1668 y(36,32,160,327,104]\))o(;)512 1734 y Fn(>)582 1738 y
Fo(B)f(:=)h(vector\([17,4,55,)o(89]\);)512 1816 y Fp(a\))10 b(Determine)g(the)g(rank)g(of)f(A.)h(F)m(rom)e(this)i(information,)d (determine)j(ho)o(w)g(man)o(y)450 1866 y(free)15 b(v)n(ariables)e(the)h (system)g(A)g Fk(X)g Fp(=)g Fk(0)g Fp(will)e(ha)o(v)o(e.)512 1916 y(b\))i(Ho)o(w)g(man)o(y)e(spanning)h(v)o(ectors)i(will)e(the)h(n)o(ull) f(space)i(of)e(A)h(con)o(tain?)512 1966 y(c\))h(Using)f(Maple's)g Fk(n)o(ullspace)d Fp(command,)g(\014nd)j(a)g(spanning)g(set)h(for)f(the)h(n)o (ull)450 2016 y(space)k(of)f(A.)g(Since)g(this)h(command)c(returns)20 b(a)e(set)h(of)e(v)o(ectors,)j(extract)f(all)e(the)450 2066 y(v)o(ectors)e(from)d(this)i(set,)g(naming)e(them)h(w1,)g(w2,)g(etc.)512 2132 y Fn(>)582 2136 y Fo(q)21 b(:=)h(nullspace\(A\);)512 2202 y Fn(>)582 2206 y Fo(for)f(k)h(from)f(1)g(to)g(4)h(do)f(w.k)h(:=)f(q[k];)g (od;)512 2284 y Fp(d\))e(By)f(insp)q(ection,)h(\014nd)f(a)g(v)o(ector)h Fk(X)f Fp(satisfying)f(the)i(equation)f(A)g Fk(X)g Fp(=)h Fk(B)p Fp(.)450 2334 y(V)m(erify)13 b(that)h(y)o(our)g(guess)h(indeed)f(satis\014es) h(the)g(equation.)512 2384 y(e\))h(If)g Fk(F)f Fp(is)h(a)f(general)h(linear)f (com)o(bination)e(of)i(the)h(v)o(ectors)h Fk(w1)p Fp(,)f Fk(w2)p Fp(,)g(...,)e Fk(w4)p Fp(,)450 2434 y(sho)o(w)g(that)g Fk(C)g Fp(=)h Fk(X)f Fp(+)g Fk(F)g Fp(is)g(still)f(a)h(solution)f(to)h(the)g (equation)g(A)g Fk(X)g Fp(=)g Fk(B)p Fp(.)g(\(Note:)p eop bop 150 275 a Fm(1.5)13 b(Maple)h(On)g(Line)967 b(25)150 391 y Fp(Y)m(ou)14 b(migh)o(t)e(need)j(Maple's)f Fk(prin)o(t)e Fp(command)g(as)i(w)o(ell)g(as)g Fk(ev)m(alm)g Fp(to)g(force)h(Maple)150 441 y(to)f(displa)o(y)f(the)h(results)h(of)e(y)o(our)h(computations.\))212 491 y(f)s(\))d(Explain)e(the)j(statemen)o(t)e("The)h(general)g(solution)f(to) g(A)h Fk(X)f Fp(=)h Fk(B)g Fp(is)f(the)h(v)o(ector)150 541 y Fk(Xo)k Fp(+)g(W)f Fk(Y)p Fp(,)h(where)h(W)e(is)h(the)h(matrix)d(whose)i (columns)f(are)h Fk(w1)p Fp(,)g Fk(w2)p Fp(,)g Fk(w3)p Fp(,)g(and)150 591 y Fk(w4)p Fp(,)e(and)f Fk(Y)h Fp(is)f(an)o(y)g(v)o(ector)h(in)f Fj(R)680 576 y Fi(4)699 591 y Fp(.")f(Hin)o(t:)17 b(Computationally)m(,)9 b(it)j(w)o(ould)g(b)q(e)h(useful)150 640 y(to)f(form)e(the)j(matrix)d(W)i (with)g(the)g(augmen)o(t)f(command,)e(form)i(the)h(v)o(ector)h Fk(Y)f Fp(with)150 690 y(four)g(parameters)h(for)f(comp)q(onen)o(ts,)f(and)i (to)f(\014nd)g(the)h(general)g(solution)f(to)g(A)g Fk(X)h Fp(=)150 740 y Fk(B)h Fp(via)f(the)h(linsolv)o(e)f(command.)i(This)f(solution)f (should)h(matc)o(h)f Fk(Xo)g Fp(+)i(W)e Fk(Y)p Fp(.)212 790 y(g\))i(Find)g(a)f(basis)h(for)g(the)g(n)o(ull)f(space)i(of)e(A)h(b)o(y)g (solving)f(the)h(equation)g(A)g Fk(X)g Fp(=)150 840 y Fk(0)j Fp(for)f(the)i(general)f(solution,)f Fk(X)p Fp(.)g(This)h(is)g(easily)f(done) h(in)g(Maple)f(b)o(y)h(using)f(the)150 889 y Fk(linsolv)o(e)10 b Fp(command.)15 b(This)d(command)f(tak)o(es)i(as)g(argumen)o(ts,)f(the)h (matrix)f(A,)g(and)150 939 y(a)i(v)o(ector)i(\(or)f(list\))f(of)g(zeros)i(as) f(the)h(righ)o(t-hand)e(side)h(v)n(alues.)20 b(Maple)15 b(will)e(deliv)o(er) 150 989 y(a)g(linear)f(com)o(bination)e(of)j(the)g(v)o(ectors)h Fk(w1)p Fp(,)f Fk(w2)p Fp(,)g(...,)e Fk(w4)i Fp(that)g(w)o(ere)h(found)f(b)o (y)f(the)150 1039 y Fk(n)o(ullspace)f Fp(command.)212 1089 y(h\))k(Find)g(a)f(basis)h(for)f(the)h(n)o(ull)f(space)i(of)e(A.)g(this)h (time)e(solving)h(the)h(system)g(A)150 1139 y Fk(X)i Fp(=)g Fk(0)g Fp(b)o(y)g(augmen)o(ting)d(A)j(with)g(a)g(column)e(of)h(zeros)j(and)d (using)h(the)g Fk(rref)f Fp(and)150 1188 y Fk(bac)o(ksub)10 b Fp(commands.)k(The)e(solution)e(will)f(not)i(b)q(e)g(readily)g(recognized)h (as)f(a)f(linear)150 1238 y(com)o(bination)h(of)j(the)g(v)o(ectors)h Fk(w1)p Fp(,)f Fk(w2)p Fp(,)g(...,)e Fk(w4)p Fp(.)212 1288 y(i\))h(V)m(erify)f(that)g(the)i(basis)e(found)h(in)f(part)h(\(h\))f(is)h (equiv)n(alen)o(t)f(to)g(the)i(basis)e Fg(f)p Fk(w1)p Fp(,)150 1338 y Fk(w2)p Fp(,)h(...,)f Fk(w4)p Fg(g)p Fp(,)h(sho)o(w)g(that)h(the)g (set)g(of)f(equations)g Fl(a)h Fk(w1)f Fp(+)h Fl(b)f Fk(w2)h Fp(+)f Fl(c)h Fk(w3)g Fp(+)f Fl(d)h Fk(w4)150 1388 y Fp(=)f Fj(v)215 1394 y Fn(k)249 1388 y Fp(has)g(a)g(solution)f(for)h(eac)o(h)g
Fj(v)687 1394 y Fn(k)721 1388 y Fp(in)f(the)i(basis)f(found)g(in)f(part)h (\(h\).)18 b(This)13 b(is)g(most)150 1437 y(easily)h(done)h(b)o(y)g(augmen)o (ting)d(W)i(with)h(the)g(general)g(solution)f(found)g(in)g(part)h(\(h\),)150 1487 y(and)g(using)h(the)g(rref)g(command)d(to)j(sho)o(w)f(the)i(equations)e (are)h(consisten)o(t)h(for)e(an)o(y)150 1537 y(v)n(alues)f(of)f(the)h (parameters)g(in)g(that)g(general)g(solution.)p eop bop 450 275 a Fp(26)967 b Fm(1.5)13 b(Maple)h(On)g(Line)p eop bop 150 703 a Fs(Chapter)31 b(2)150 911 y Fr(Dimension)150 1153 y Fq(2.1)66 b(On)22 b(Line)150 1245 y Fp(Restart)f(Maple)g(to)g(clear)g (its)g(memory)d(of)i(all)g(v)n(ariables,)h(and)f(re-initialize)g(b)o(y)150 1295 y(loading)12 b(the)j Fl(linalg)e Fp(pac)o(k)n(age.)150 1414 y Fe(Exercise)k(1)150 1491 y Fp(Giv)o(en)c(the)i(matrix)d(A)i(en)o (tered)h(b)q(elo)o(w,)212 1558 y Fn(>)282 1562 y Fo(A)21 b(:=)h (matrix\(4,3,[1,2,)o(-3,4,)o(5,-1,)o(3,2,1)o(,1,1,)o(1]\);)212 1641 y Fp(\014nd)17 b(the)g(reduced)h(\(ro)o(w\))e(ec)o(helon)h(form)d(of)i (A.)g(Ho)o(w)g(can)g(y)o(ou)g(tell)g(just)h(from)150 1691 y(this)i(reduced)h (form)d(that)i(the)g(columns)e(of)h(A)h(are)g(indep)q(enden)o(t?)34 b(Relate)18 b(y)o(our)150 1741 y(answ)o(er)d(to)e(Theorem)h(1.)150 1860 y Fe(Exercise)j(2)150 1938 y Fp(Let)f(A)h(b)q(e)f(a)g(\(random\))f (matrix)f(with)h(more)g(ro)o(ws)i(than)f(columns.)23 b(State)16 b(a)g(gen-)150 1988 y(eral)g(rule)g(for)g(using)g(rref\(A\))h(to)f(decide)h (whether)g(or)f(not)g(the)h(columns)e(of)g(A)i(are)150 2037 y(indep)q(enden)o(t.)i(Demonstrate)13 b(y)o(our)g(condition)g(b)o(y)g(\(a\))h (pro)q(ducing)f(a)g(5)g(x)h(4)f(matrix)150 2087 y(A)18 b(with)g(no)g (non-zero)h(en)o(tries,)h(and)e(with)g(indep)q(enden)o(t)h(columns;)g(and)f (b)o(y)g(\(b\),)150 2137 y(pro)q(ducing)g(a)g(5)g(x)g(4)g(matrix)f(with)h(no) g(non-zero)g(en)o(tries,)i(and)e(with)g(dep)q(enden)o(t)150 2187 y(columns.)e(In)c(eac)o(h)g(case,)h(obtain)e(rref\(A\).)i(Pro)o(v)o(e)f (y)o(our)g(condition)f(using)g(Theorem)150 2237 y(1.)150 2356 y Fe(Exercise)17 b(3)150 2434 y Fp(Let)d(A)g(b)q(e)h(the)f(matrix)e(en)o (tered)k(b)q(elo)o(w.)804 2539 y(27)p eop bop 450 275 a Fp(28)967 b Fm(2.2)13 b(Maple)h(On)g(Line)512 387 y Fn(>)582 391 y Fo(A)21 b(:=)h(matrix\(5,6,[-1,2)o(,6,-8)o(,-14,)o (3,2,4)o(,1,-8)o(,5,-)o(1,-3,)o(1,4,-)o(9,-10)o(,0,3,)o(-2,-1)o(,12,)o(-)512 434 y(1,4,5,7,11,-11,-19,)o(9]\);)512 528 y Fp(Use)f(the)g Fk(rref)d Fp(command)g(to)i(\014nd)g(the)g(piv)o(ot)f(columns)g(of)h(A.)f(W)m (rite)h(them)450 578 y(explicitly)13 b(as)h(columns.)j(Then)d(express)i(the)e (other)h(columns)e(of)g(A)h(as)g(linear)f(com-)450 628 y(binations)i(of)g (the)h(piv)o(ot)f(columns.)22 b(\(See)17 b(Example)d(4)h(in)h(the)g(text.\)) 24 b(Y)m(ou)15 b(should)450 678 y(disco)o(v)o(er)f(that)g(the)h(\014rst)f (three)i(columns)c(of)i(A)g(are)g(the)g(piv)o(ot)f(columns.)450 818 y Fe(Exercise)k(4)450 903 y Fp(If)i Fj(A)528 909 y Fn(k)568 903 y Fp(represen)o(ts)j(the)d(kth)h(column)d(of)i(the)h(matrix)d(A)j (de\014ned)g(in)f(Exercise)i(3,)450 953 y(form)15 b(the)j(matrix)d(B)j(whose) f(columns)f(are)i(the)g(columns)d(of)i(A)g(in)g(the)h(follo)o(wing)450 1003 y(order:)k(B)16 b(=)g([)f Fj(A)729 1009 y Fi(4)748 1003 y Fj(;)e(A)804 1009 y Fi(6)823 1003 y Fj(;)g(A)879 1009 y Fi(1)898 1003 y Fj(;)g(A)954 1009 y Fi(2)973 1003 y Fj(;)g(A)1029 1009 y Fi(3)1048 1003 y Fj(;)g(A)1104 1009 y Fi(5)1123 1003 y Fp(].)23 b(\(This)15 b(is)h(most)e(easily)h(done)h(b)o(y)f(using)450 1053 y(the)h Fk(augmen)o(t)c Fp(and)j Fk(col)f Fp(commands.\))19 b(Find)c(the)g(piv)o(ot)f(columns)g(of)g(B)i(b)o(y)e(using)450 1103 y(the)j Fk(rref)e Fp(command.)23 b(Do)16 b(y)o(ou)g(obtain)g(a)g (di\013eren)o(t)i(set)f(of)f(piv)o(ot)f(columns?)25 b(Use)450 1152 y(rref\(B\))12 b(to)e(express)i(the)f(other)g(columns)e(of)g(B)i(as)f (linear)g(com)o(binations)e(of)i(the)h(piv)o(ot)450 1202 y(columns.)18 b(Could)13 b(y)o(ou)h(ha)o(v)o(e)g(deriv)o(ed)g(these)i(expressions)f(from)e (those)i(in)e(Exercise)450 1252 y(3?)18 b(If)13 b(so,)h(ho)o(w?)450 1392 y Fe(Exercise)j(5)450 1477 y Fp(Find)h(a)f(matrix)g(C)g(whose)i(columns)
e(are)h(just)g(those)h(of)f(A)g(listed)g(in)f(a)h(di\013eren)o(t)450 1527 y(order,)f(suc)o(h)f(that)h(the)f(column)e(of)i(C)g(whic)o(h)g(equals)g Fj(A)1357 1533 y Fi(5)1391 1527 y Fp(and)g(the)h(column)d(whic)o(h)450 1577 y(equals)f Fj(A)606 1583 y Fi(1)638 1577 y Fp(are)h(b)q(oth)f(piv)o(ot)g (columns.)j(Is)e(it)f(p)q(ossible)g(to)g(\014nd)h(suc)o(h)g(a)f(C)g(where)h Fj(A)1781 1583 y Fi(2)450 1627 y Fp(is)g(a)g(piv)o(ot)g(column)f(as)h(w)o (ell?)20 b(If)14 b(so,)g(\014nd)g(an)h(example.)j(If)c(not,)g(explain)g(wh)o (y)g(it)g(is)450 1677 y(not)g(p)q(ossible.)450 1838 y Fq(2.2)66 b(On)22 b(Line)450 1937 y Fp(Restart)12 b(Maple)g(to)f(clear)h(its)g(memory)d (of)i(all)g(v)n(ariables,)f(and)i(re-initialize)f(b)o(y)g(load-)450 1987 y(ing)j(the)h Fl(linalg)f Fp(pac)o(k)n(age.)19 b(In)c(addition,)e(use)i (the)g(command)d(")i Fo(with\(student\):)p Fp(")450 2037 y(to)g(load)f(the)h Fl(student)g Fp(pac)o(k)n(age)g(in)f(order)i(to)f(access)h(its)f Fk(equate)f Fp(command.)450 2177 y Fe(Exercise)k(1)450 2262 y Fp(Let)d(A)g(b)q(e)h(the)f(matrix)e(en)o(tered)k(b)q(elo)o(w.)512 2341 y Fn(>)582 2345 y Fo(A)21 b(:=)h(matrix\(5,6,[-1,2)o(,6,-8)o(,-14,)o (3,2,4)o(,1,-8)o(,5,-)o(1,-3,)o(1,4,-)o(9,-10)o(,0,3,)o(-2,-1)o(,12,)o(-)512 2387 y(1,4,5,7,11,-11,-19,)o(9]\);)p eop bop 150 275 a Fm(2.2)13 b(Maple)h(On)g(Line)967 b(29)150 391 y Fk(P)o(art)15 b(\(a\))150 470 y Fp(Find)i(the)g(rank)g(of)f(A)h(via)f(the)h Fk(rank)g Fp(command.)24 b(Using)16 b(only)g(the)i(v)n(alue)e(of)g(the)150 520 y(rank,)d(explain)g(wh)o(y)h(the)h(statemen)o(ts)f Fl(i)p Fp(\))f(and)h Fl(ii)p Fp(\))g(giv)o(en)f(b)q(elo)o(w)h(are)g(true.)212 571 y Fl(i)p Fp(\))g(The)h(reduced)h(form)c(of)i(the)h(augmen)o(ted)e(matrix) f(for)i(the)h(system)f(A)g Fk(X)h Fp(=)f Fk(0)150 620 y Fp(has)h(three)h (free)g(v)n(ariables.)k(\(Recall)14 b(that)h(in)f(a)h(previous)g(On)g(Line)g (section)h(it)e(w)o(as)150 670 y(noted)g(that)g(the)h(rank)f(is)f(the)i(n)o (um)o(b)q(er)e(of)g(non-zero)i(ro)o(ws)f(in)f(the)i(reduced)g(form.\))212 721 y Fl(ii)p Fp(\))i(The)g(n)o(ull)f(space)i(of)e(A)h(has)g(dimension)e(3.) 27 b(\(Hin)o(t:)d(Ho)o(w)16 b(man)o(y)f(spanning)150 771 y(v)o(ectors)g(are)f (there)i(in)d(the)h(general)h(solution)e(to)g(A)h Fk(X)g Fp(=)g Fk(0)p Fp(?\))150 885 y Fk(P)o(art)h(\(b\))150 963 y Fp(Sho)o(w)f(that)g(eac) o(h)h(of)e(the)i(v)o(ectors)g Fk(X1)p Fp(,)f Fk(X2)p Fp(,and)f Fk(X3)h Fp(giv)o(en)g(b)q(elo)o(w)g(satisfy)f(A)i Fk(X)f Fp(=)150 1013 y Fk(0)p Fp(.)212 1081 y Fn(>)282 1085 y Fo(X1)21 b(:=)h (vector\([-5,13,-)o(10,2,)o(-3,1])o(\);)d(X2)i(:=)h(vector\([3,-6,11)o(,1,2,) o(-5]\);)212 1128 y(X3)g(:=)f(vector\([-4,7,9,5,1)o(,-6])o(\);)212 1209 y Fp(Note:)d(Since)13 b(v)o(erifying)f(that)h(A)f Fj(X)771 1215 y Fn(k)805 1209 y Fp(=)h Fk(0)f Fp(is)h(a)f(rep)q(etitiv)o(e)i(task,)e (it)g(can)h(b)q(e)h(done)150 1258 y(in)f(a)h Fl(for-lo)n(op)p Fp(.)150 1372 y Fk(P)o(art)h(\(c\))150 1450 y Fp(By)h(computing)d(the)j(rank) g(of)e(the)i(matrix)e([)p Fk(X1)p Fp(,)h Fk(X2)p Fp(,)g Fk(X3)p Fp(],)f(pro)o(v)o(e)i(that)f Fk(X1)p Fp(,)h Fk(X2)p Fp(,)150 1500 y(and)j Fk(X3)h Fp(are)g(linearly)f(indep)q(enden)o(t.)36 b(\(Recall)19 b(that)h(the)g(maxima)o(l)d(n)o(um)o(b)q(er)i(of)150 1550 y(linearly)13 b(indep)q(enden)o(t)i(columns)e(equals)h(the)g(rank.\))150 1664 y Fk(P)o(art)h(\(d\))150 1742 y Fp(Ho)o(w)i(do)q(es)h(it)f(follo)o(w)e (that)j(the)f(dimension)f(of)h(the)h(n)o(ull)e(space)i(of)f(A)g(is)g(3?)28 b(Ho)o(w)150 1792 y(do)q(es)15 b(it)e(follo)o(w)f(that)i(the)212 1843 y Fj(X)246 1849 y Fn(k)283 1843 y Fp(constitute)h(a)f(basis)f(for)h(the) g(n)o(ull)f(space?)150 1957 y Fk(P)o(art)i(\(e\))150 2035 y Fp(Using)i(Maple's)f Fk(n)o(ullspace)e Fp(command,)h(\014nd)i(a)g(basis)g (for)f(the)i(n)o(ull)e(space)i(of)e(A.)150 2085 y(Express)g(eac)o(h)f(v)o (ector)h(in)e(this)g(basis)h(as)g(a)f(linear)g(com)o(bination)e(of)i(the)h Fj(X)1352 2091 y Fn(k)1373 2085 y Fp('s)g(from)150 2135 y(part)g(\(d\).)20 b(Hin)o(t:)f(Giv)o(en)14 b(t)o(w)o(o)g(bases)h(for)g(this)f(n)o(ull)g(space,) h(sho)o(wing)f(that)g(they)i(are)150 2185 y(equiv)n(alen)o(t)c(requires)i (sho)o(wing)e(that)h(an)o(y)f(v)o(ector)h(in)g(one)g(can)g(b)q(e)g(found)f (as)h(a)f(linear)150 2234 y(com)o(bination)i(of)h(the)i(v)o(ectors)h(in)e
(the)h(other.)25 b(Hence,)18 b(a)e(set)i(of)d(equations)i(of)e(the)150 2284 y(form)d(a)i(X1)g(+)g(b)g(X2)g(+)g(c)g(X3)g(=)g Fj(w)718 2290 y Fn(k)752 2284 y Fp(m)o(ust)f(b)q(e)i(solv)o(ed)f(for)f(eac)o(h)i(k)e (=)i(1,)e(2,)g(3.)18 b(This)150 2334 y(can)13 b(b)q(e)g(done)g(b)o(y)f (forming)f(the)i(augmen)o(ted)f(matrix)e([)p Fk(X1)p Fp(,)i Fk(X2)p Fp(,)g Fk(X3)p Fp(,)h Fk(w1)p Fp(,)f Fk(w2)p Fp(,)h Fk(w3)p Fp(])150 2384 y(and)j(ro)o(w)f(reducing.)24 b(In)16 b(ro)o(w)g(reduced)h(form)d(this)i(matrix)e(will)g(indicate)i(whether)150 2434 y(or)d(not)g(these)h(equations)f(are)h(solv)n(able,)d(and)i(if)f(so,)h (ho)o(w)g(to)g(express)h(the)g(non-piv)o(ot)p eop bop 450 275 a Fp(30)967 b Fm(2.3)13 b(Maple)h(On)g(Line)450 391 y Fp(v)o(ectors)19 b(in)e(terms)g(of)h(the)g(piv)o(ot)f(v)o(ectors.)30 b(See)19 b(Theorem)e(1)g(and)h(Example)e(4)h(in)450 441 y(Section)d(2.1.)450 546 y Fk(P)o(art)h(\(f)5 b(\))450 622 y Fp(In)12 b(part)h(\(e\),)f(what)g (made)f(us)i(so)f(sure)i(that)e(the)h(\014rst)g(three)g(columns)e(w)o(ould)h (b)q(e)g(the)450 672 y(piv)o(ot)h(columns?)18 b(Wh)o(y)c(couldn't,)f(for)h (example,)e(the)j(piv)o(ot)e(columns)g(b)q(e)i(columns)450 722 y(1,)i(3,)g(and)g(4?)27 b(\(Hin)o(t:)d(Think)16 b(ab)q(out)h(what)g(this) g(w)o(ould)f(imply)f(for)i(the)g(reduced)450 772 y(form)12 b(of)h([)p Fk(X1)p Fp(,)g Fk(X2)p Fp(,)h Fk(X3)p Fp(].\))450 876 y Fk(P)o(art)h(\(g\))450 953 y Fp(In)i(part)h(\(e\))g(y)o(ou)f(expressed) j(the)e(v)o(ectors)g Fj(w)1188 959 y Fn(k)1225 953 y Fp(in)f(terms)g(of)g (the)h(v)o(ectors)h Fj(X)1701 959 y Fn(k)1722 953 y Fp(.)28 b(In)450 1003 y(this)13 b(part,)f(no)o(w)h(express)h(the)g Fj(X)964 1009 y Fn(k)997 1003 y Fp(in)e(terms)h(of)f(the)h Fj(w)1305 1009 y Fn(k)1325 1003 y Fp(.)18 b(This)12 b(w)o(ould)g(complete)g (the)450 1053 y(demonstration)f(that)i(the)g Fj(X)916 1059 y Fn(k)950 1053 y Fp(and)f(the)h Fj(w)1129 1059 y Fn(k)1162 1053 y Fp(are)g(equiv)n(alen)o(t)f(spanning)g(sets)i(for)e(the)450 1102 y(n)o(ull)h(space)i(of)e(A.)h(Hin)o(t:)j(Use)e(the)g(tec)o(hnique)f(in)g (part)g(\(e\).)450 1207 y Fk(P)o(art)h(\(h\))450 1383 y Fp(Find)e(\(b)o(y)g (insp)q(ection\))g(a)g(v)o(ector)h Fk(T)f Fp(whic)o(h)g(solv)o(es)g(the)h (equation)f(A)g Fk(X)g Fp(=)1650 1250 y Ff(2)1650 1323 y(6)1650 1348 y(6)1650 1373 y(6)1650 1398 y(6)1650 1424 1283 y(4)1719 y Fp(6)1719 1333 y(1)1719 1383 y(4)1719 1433 y(1)1699 1482 y(11)1761 1250 y Ff(3)1761 1323 y(7)1761 1348 y(7)1761 1373 y(7)1761 1398 y(7)1761 1424 y(5)1788 1383 y Fp(.)450 1587 y Fk(P)o(art)i(\(i\))450 1714 y Fp(Let)526 1630 y Ff(2)526 1705 y(4)574 1664 y Fj(r)574 1713 y(s)577 1763 y(t)615 1630 y Ff(3)615 1705 y(5)658 1714 y Fp(b)q(e)h(an)f(arbitrary)g(elemen)o(t)g(of)g Fj(R)1189 1699 y Fi(3)1222 1714 y Fp(and)h(let)f Fk(Z)g Fp(=)h Fk(T)f Fp(+)h(r)g Fk(X1)f Fp(+)g(s)h Fk(X2)450 1814 y Fp(+)e(t)g Fk(X3)p Fp(,)g(where)h Fk(T)f Fp(is)g(as)g(found)g(in)g(part)g(\(h\).)19 b(Compute)13 b(A)h Fk(Z)p Fp(.)g(Explain)f(wh)o(y)g(y)o(ou)450 1863 y(get)h(what)g(y)o(ou)g(get.)19 b(Find)13 b(constan)o(ts)i(u,)f(v,)g (and)f(w)h(suc)o(h)h(that)f Fk(Z)h Fp(=)f Fk(T)g Fp(+)g(u)g Fk(w1)h Fp(+)450 1913 y(v)f Fk(w2)g Fp(+)g(w)g Fk(w3)p Fp(.)k(What)c(theorem) f(do)q(es)i(this)f(demonstrate?)512 1963 y(Note:)k(An)12 b(e\016cien)o(t)g (ea)o(y)g(of)f(doing)g(this)h(is)g(to)f(use)i(the)g Fk(equate)e Fp(command)e(from)450 2013 y(Maple.)17 b(One)c(migh)o(t)c(\014rst)k(en)o(ter) g(the)f(t)o(w)o(o)g(prop)q(osed)g(expressions)i(for)d Fk(Z)h Fp(as)g(follo)o(ws:)512 2078 y Fn(>)582 2082 y Fo(q1)21 b(:=)h (evalm\(augment\(X)o(.\(1..)o(3\)\)&*)o(vecto)o(r\([r,)o(s,t])o(\)\);)d(q2)i (:=)g(evalm\(augment\(w.\(1.)o(.3\)\)&)o(*vect)o(or\([u)o(,v,w])o(\)\);)512 2159 y Fp(W)m(e)16 b(can)f(equate)i(these)g(t)o(w)o(o)e(v)o(ectors)i(with)e (the)i Fk(equate)e Fp(command)d(from)i(the)450 2208 y Fl(student)c Fp(pac)o(k)n(age)f(and)g(then)h(solv)o(e)g(for)f(the)h(required)g(constan)o (ts)g(using)g(the)g(follo)o(wing)450 2258 y(syn)o(tax.)512 2323 y Fn(>)582 2327 y Fo(q3)21 b(:=)h(equate\(q1,q2\);)512
2392 y Fn(>)582 2396 y Fo(q4)f(:=)h(solve\(q3,)p Fg(f)p Fo(u,v,w)o Fg(g)p Fo(\);)d(q5)i(:=)g(solve\(q3,)p Fg(f)p Fo(r,s,t)p Fg(g)p Fo(\);)p eop bop 150 275 a Fm(2.3)13 b(Maple)h(On)g(Line)967 b(31)150 391 y Fq(2.3)66 b(On)22 b(Line)150 484 y Fp(Restart)13 b(Maple)f(to)h(clear)f (all)g(v)n(ariables)f(and)h(reset)j(its)d(memory)m(,)d(then)k(initialize)e(b) o(y)150 534 y(loading)h(the)j Fl(linalg)e Fp(pac)o(k)n(age.)150 656 y Fe(Exercise)k(1)150 735 y Fp(Using)10 b(Maple's)g Fk(randmatrix)d Fp(command,)h(construct)k(M,)d(a)h(random)f(3)h(x)f(5)h(matrix.)150 785 y(What)k(do)g(y)o(ou)g(exp)q(ect)h(for)f(the)h(rank)f(of)g(M?)g(Chec)o (k,)h(using)f(the)g Fk(rank)g Fp(command.)150 834 y(Is)g(it)f(conceiv)n(able) g(that)h(the)g(rank)g(could)f(ha)o(v)o(e)g(turned)i(out)e(otherwise?)19 b(Wh)o(y)13 b(is)g(it)150 884 y(unlik)o(ely?)25 b(Finally)m(,)14 b(retain)j(this)f(matrix)f(for)h(use)h(in)f(the)h(other)g(exercises)i(of)d (this)150 934 y(section.)150 1056 y Fe(Exercise)h(2)150 1135 y Fp(F)m(orm)9 b(t)o(w)o(o)i(di\013eren)o(t)h(random)d(linear)i(com)o (binations)e(of)h(the)i(three)g(ro)o(ws)f(of)g(M,)f(then)150 1184 y(app)q(end)16 b(these)g(t)o(w)o(o)f(ro)o(ws)g(to)g(M,)g(thereb)o(y)h (creating)f(a)g(5)g(x)g(5)g(matrix.)k(It)d(helps)f(to)150 1234 y(use)h Fk(rand)f Fp(to)g(de\014ne)h(a)g(function)f(f)g(as)g(a)g(generator)i (for)e(the)h(random)e(co)q(e\016cien)o(ts)150 1284 y(needed)19 b(for)d(the)i(linear)f(com)o(binations.)25 b(Then,)18 b(the)g Fk(sum)e Fp(and)h Fk(ro)o(w)f Fp(commands)150 1334 y(simplify)9 b(constructing)14 b(the)f(linear)f(com)o(binations)e(of)h(the)i(ro)o(ws)g(of) e(M.)h(Finally)m(,)e(the)150 1384 y Fk(stac)o(k)16 b Fp(command)e(app)q(ends) j(ro)o(ws)g(to)f(the)h(b)q(ottom)e(of)h(M.)g(\(See)h(Exercise)i(3)d(from)150 1434 y(the)j(On)g(Line)f(exercises)j(for)d(Section)h(1.4.\))31 b(Using)19 b(b)q(oth)f Fk(rank)h Fp(and)f Fk(rref)p Fp(,)g(test)150 1483 y(the)d(rank)g(of)g(the)g(enlarged)g(matrix.)k(What)c(is)g(the)g(maxim)o (al)d(n)o(um)o(b)q(er)i(of)g(linearly)150 1533 y(indep)q(enden)o(t)h(columns) e(in)g(the)i(new)f(matrix?)150 1655 y Fe(Exercise)j(3)150 1734 y Fp(F)m(or)h(the)g(5)g(x)g(5)g(matrix)e(created)k(in)e(Exercise)h(2,)g (\014nd)f(a)g(set)h(of)e(columns)g(whic)o(h)150 1784 y(forms)f(a)g(basis)h (for)g(the)g(column)f(space.)28 b(Express)19 b(the)e(other)h(columns)d(as)i (linear)150 1833 y(com)o(binations)11 b(of)i(these)h(columns.)j(Use)d(the)g (tec)o(hnique)g(of)f(Example)f(3)h(in)g(Section)150 1883 y(2.1.)150 2005 y Fe(Exercise)k(4)150 2084 y Fp(In)e(this)f(exercise)i(y)o(ou)f(will)e (explore)i(Maple's)f(abilit)o(y)f(to)h(obtain)g(the)h(reduced)h(ro)o(w)150 2134 y(ec)o(helon)c(form)d(n)o(umerically)m(.)14 b(In)d(Exercise)i(3)d(the)i (reduction)g(is)e(found)h(sym)o(b)q(olically)l(,)150 2184 y(using)e(exact)i (arithmetic,)d(and)i(the)g(result)g(su\013ers)i(no)d(loss)h(of)f(accuracy)h (from)e(round-)150 2233 y(o\013)14 b(error.)20 b(A)14 b(\015oating)f(p)q(oin) o(t)h(ev)n(aluation)f(of)g(this)h(exact)h(answ)o(er)g(will)e(serv)o(e)i(as)f (the)150 2283 y(target)g(answ)o(er)h(that)f(w)o(e)g(will)e(exp)q(ect)k(Maple) e(to)f(deliv)o(er)h(n)o(umerically)m(.)212 2334 y(First,)22 b(apply)e(the)h Fk(ev)m(alf)f Fp(command)d(to)k(the)g(reduced)h(ro)o(w)e(ec)o (helon)h(matrix)150 2384 y(found)14 b(in)f(Exercise)j(3.)i(Next,)c(b)o(y)f (mapping)f(the)j(con)o(v)o(ert-to-\015oat)f(op)q(eration)f(on)o(to)150 2434 y(it,)20 b(con)o(v)o(ert)h(the)f(5)f(x)h(5)f(matrix)f(of)h(Exercise)i(2) f(to)f(\015oating)g(p)q(oin)o(t)g(form.)34 b(\(See)p eop bop 450 275 a Fp(32)967 b Fm(2.3)13 b(Maple)h(On)g(Line)450 391 y Fp(Exercise)21 b(1)f(in)f(the)i(On)f(Line)g(exercises)i(for)d(Section)i (1.4.\))35 b(Then,)21 b(obtain)e(the)450 441 y(reduced)d(ro)o(w)f(ec)o(helon) g(form)e(of)h(this)h(n)o(umeric)f(matrix.)19 b(Observ)o(e)d(that)f(the)g (result)450 491 y(is)h(wildly)e(wrong.)23 b(In)16 b(fact,)g(it)f(sho)o(ws)h (the)g(matrix)e(to)i(b)q(e)g(of)f(rank)h(4)f(whereas)i(the)450 541 y(matrix)12 b(is)i(kno)o(wn)f(to)h(ha)o(v)o(e)g(rank)f(3.)512 606 y Fn(>)582 610 y Fo(mm)21 b(:=)h(map\(convert,MM,)o(float)o(\);)512 675 y Fn(>)582 679 y Fo(mm1:=rref\(mm\);)512 756 y Fp(The)e(reason)f(for)f
(the)i(error)g(can)f(b)q(e)g(seen)h(b)o(y)f(ro)o(w)f(reducing)i(the)f(matrix) e(to)450 806 y(an)d(upp)q(er-triangular)g(form,)e(without)i(dividing)e(the)j (diagonal)d(elemen)o(ts)i(to)g(mak)o(e)450 855 y(them)h(1's.)22 b(This)15 b(will)f(prev)o(en)o(t)i(division)f(b)o(y)g(p)q(ossible)g(small)f (n)o(um)o(b)q(ers.)22 b(This)15 b(ro)o(w)450 905 y(reduction)g(can)f(b)q(e)g (accomplished)f(via)g(the)i Fk(gausselim)c Fp(command.)512 970 y Fn(>)582 974 y Fo(q)21 b(:=)h(gausselim\(mm\);)512 1051 y Fp(So,)d(although)f(the)h(w)o(ork)g(w)o(as)f(done)h(n)o(umerically)m(,)e (the)i(only)f(di\016cult)o(y)g(that)450 1101 y(has)c(surfaced)h(is)f(the)h (small)d(p)q(ositiv)o(e)h(v)n(alue)h(on)f(the)i(main)d(diagonal,)g(a)i(v)n (alue)f(that)450 1151 y(in)20 b(exact)g(arithmetic)f(w)o(ould)g(b)q(e)i (zero.)37 b(T)m(o)19 b(get)h(Maple)g(to)g(render)h(suc)o(h)g(small)450 1200 y(v)n(alues)11 b(as)g(zeros,)h(map)d(the)j Fk(fnormal)c Fp(command)h(on)o(to)h(the)i(matrix.)j(The)c Fk(fnormal)450 1250 y Fp(command)g(tak)o(es)k(as)f(additional)e(parameter,)h(an)h(in)o (teger)g(sp)q(ecifying)g(the)h(n)o(um)o(b)q(er)450 1300 y(of)e(digits)h(to)f (whic)o(h)h(the)h(rounding)e(is)h(to)f(b)q(e)i(p)q(erformed.)512 1365 y Fn(>)582 1369 y Fo(rref\(map\(fnormal)o(,q,9\))o(\);)512 1446 y Fp(This)k(result)g(no)o(w)f(matc)o(hes)f(what)i(w)o(as)f(obtained)g (when)h(the)g(exact)g Fl(solution)450 1496 y Fp(w)o(as)14 b(con)o(v)o(erted)h (to)f(\015oating)e(p)q(oin)o(t)i(form.)450 1639 y Fe(Exercise)j(5)450 1715 y Fp(Find)11 b(a)h(basis)g(for)f(the)h(column)e(space)j(of)e(the)h(5)g (x)f(5)h(of)f(Exercise)i(2)e(b)o(y)h(ro)o(w)f(reducing)450 1765 y(its)16 b(transp)q(ose)h(and)e(in)o(v)o(oking)f(the)i(Non-Zero)g(Ro)o (ws)f(Theorem.)23 b(Express)17 b(eac)o(h)g(of)450 1815 y(the)g(basis)f (columns)f(found)g(in)h(Exercise)h(3)f(as)g(linear)g(com)o(binations)e(of)h (the)i(basis)450 1865 y(columns)c(found)g(b)o(y)h(this)g(tec)o(hnique.)450 2008 y Fe(Exercise)j(6)450 2084 y Fp(Create)d(four)e(random)f(4)h(x)h(4)f (matrices)g(of)g(rank)h(1,)f(2,)g(3,)g(and)g(4,)g(resp)q(ectiv)o(ely)m(.)19 b(\(See)450 2134 y(Exercise)d(3)d(from)f(the)j(On)f(Line)g(exercises)i(for)d (Section)i(1.4.\))i(Determine)c(the)i(n)o(ull)450 2184 y(space)g(and)e(the)h (dimension)e(of)h(the)h(n)o(ull)e(space)j(\(called)e Fl(nul)r(lity)g Fp(in)g(some)g(texts\))h(for)450 2234 y(eac)o(h)j(matrix.)22 b(Relate)16 b(the)h(rank,)f(the)h(n)o(ullit)o(y)m(,)d(and)i(the)h(n)o(um)o(b) q(er)e(of)g(ro)o(ws)i(in)e(the)450 2284 y(matrix.)p eop bop 150 706 a Fs(Chapter)31 b(3)150 917 y Fr(T)-10 b(ransformations)150 1161 y Fq(3.1)66 b(Section)23 b(3.1)f(-)h(On)f(Line)150 1259 y Fp(Restart)12 b(Maple)f(to)g(clear)h(all)e(v)n(ariables,)h(then)h(load)e (the)i Fl(linalg)f Fp(and)g Fl(plots)g Fp(pac)o(k)n(ages.)150 1309 y(The)h(exercises)i(in)d(this)h(section)g(deal)g(with)f(transformations) f(in)i(the)g(plane.)17 b(Hence,)150 1359 y(constructing)12 b(sev)o(eral)g(di\013eren)o(t)g(t)o(yp)q(es)g(of)e(graphs)i(needed)g(in)f (this)g(section)h(requires)150 1408 y(the)i Fl(plots)g Fp(pac)o(k)n(age.)150 1543 y Fe(Exercise)j(1)150 1626 y Fp(Create)h(a)f(2)f(x)h(n)g(matrix)e(F)j (whose)f Fl(n)g Fp(columns)f(are)i(the)f(co)q(ordinates)h(of)e(certain)150 1676 y(p)q(oin)o(ts)g(in)f(the)h(plane.)23 b(These)17 b(p)q(oin)o(ts)f(are)g (the)g(endp)q(oin)o(ts)g(of)f(line)g(segmen)o(ts)h(that)150 1726 y(constitute)j(a)f(letter)h(of)e(the)i(alphab)q(et.)30 b(Because)20 b(it)e(migh)o(t)e(b)q(e)j(tedious)f(to)g(con-)150 1776 y(struct)13 b(the)f(letter)h("O")e(with)h(line)f(segmen)o(ts,)g(feel)h (free)h(to)e(select)i(some)e(more)f(easily)150 1825 y(constructed)16 b(letter,)e(suc)o(h)h(as)f("F".)212 1879 y(Sk)o(etc)o(h)i(the)h(letter)f(of)f (y)o(our)g(c)o(hoice)h(on)f(a)g(piece)i(of)e(pap)q(er,)h(using)f(a)g(minim)n (um)150 1929 y(n)o(um)o(b)q(er)10 b(of)g(arcs,)i(and)f(a)f(maxim)n(um)d(n)o (um)o(b)q(er)j(of)g(line)h(segmen)o(ts.)17 b(Pic)o(k)10 b(an)h(endp)q(oin)o (t)150 1978 y(of)k(some)f(segmen)o(t)h(as)g(the)h(origin)e(and)h(assign)g(co) q(ordinates)g(to)g(eac)o(h)h(of)f(the)g(other)150 2028 y(endp)q(oin)o(ts.)j (Sk)o(etc)o(hing)c(the)h(letter)g(on)e(a)h(sheet)h(of)e(graph)h(pap)q(er)h (migh)o(t)d(mak)o(e)g(this)150 2078 y(easier.)212 2131 y(En)o(ter)g(in)o(to)e
(the)h(matrix)d(F)j(the)g(co)q(ordinates)g(of)f(the)h(endp)q(oin)o(ts)g(of)f (the)h(segmen)o(ts)150 2181 y(making)i(up)j(y)o(our)g(letter.)25 b(Eac)o(h)16 b(column)f(represen)o(ts)j(an)e(endp)q(oin)o(t.)24 b(Start)16 b(at)g(an)150 2231 y(extremit)o(y)m(,)d(and)h(use)i(con)o(tiguous) e(columns)f(to)h(represen)o(t)j(p)q(oin)o(ts)e(connected)h(b)o(y)e(a)150 2281 y(line)e(segmen)o(t.)17 b(If)12 b(y)o(our)g(letter)i(requires)f(y)o(ou)f (to)g(retrace)i(a)e(segmen)o(t)g(\(this)h(happ)q(ens)150 2331 y(with)f(the)h(arms)e(in)h(letters)h(lik)o(e)e(E)i(and)f(F\),)g(list)f(the)i (co)q(ordinates)g(of)f(an)o(y)g(endp)q(oin)o(ts)150 2380 y(in)h(the)i(order)g (in)e(whic)o(h)h(they)g(are)g(tra)o(v)o(ersed.)212 2434 y(F)m(or)g(example,)e (a)h(recognizable)i(letter)g(F)f(can)g(b)q(e)g(represen)o(ted)j(b)o(y)d(the)g (matrix)804 2539 y(33)p eop bop 450 275 a Fp(34)967 b Fm(3.1)13 b(Maple)h(On)g(Line)913 352 y Ff(\024)956 385 y Fp(0)41 b(0)h(2)f(0)g(0)1272 369 y Fi(3)p 1272 376 17 2 v 1272 399 a(2)956 435 y Fp(0)g(2)h(2)f(2)g(1)j(1)1315 352 y Ff(\025)512 532 y Fp(where)15 b(the)g(origin)e(has)h(b)q(een)h(tak)o (en)f(at)f(the)i(base)f(of)g(the)g(v)o(ertical)g(strok)o(e.)512 584 y(Sa)o(v)o(e)j(the)h(w)o(orksheet)g(in)f(whic)o(h)g(y)o(ou)f(ha)o(v)o(e)h (en)o(tered)i(y)o(our)e(matrix)e(F)i(since)h(it)450 634 y(will)12 b(b)q(e)j(used)g(in)e(the)i(remaining)c(exercises.)450 794 y Fe(Exercise)17 b(2)450 874 y Fp(Since)g(y)o(ou)g(will)e(need)j(\(and)e(w)o (an)o(t?\))27 b(to)16 b(plot)h(y)o(our)f(letter,)i(create)g(the)g(follo)o (wing)450 924 y(Maple)d(function)g(that)g(tak)o(es)h(as)f(input)g(the)h(name) e(of)h(the)h(matrix)d(of)h(y)o(our)h(letter,)450 974 y(and)d(returns)i(a)e (plot)g(of)g(the)h(letter.)18 b(The)13 b(e\013ort)g(required)h(to)e(t)o(yp)q (e)h(in)f(this)g(function)450 1023 y(will)g(more)h(than)h(pa)o(y)g(for)f (itself)h(as)g(y)o(ou)f(exp)q(erimen)o(t)h(with)f(these)j(exercises.)512 1095 y Fn(>)582 1099 y Fo(f)21 b(:=)h(x)f(->)h(pointplot\([seq\(c)o(onver)o (t\(col)o(\(x,k\))o(,lis)o(t\),k=)o(1..co)o(ldim\()o(x\)\)],)512 1141 y(style=line,)e(axes=boxed,)f(scaling=constrained)o(\):)512 1226 y Fp(Obtain)13 b(a)f(plot)g(of)g(y)o(our)g(letter)i(b)o(y)e(applying)f (the)i(function)g(f)f(to)g(the)h(v)n(ariable)f(F)450 1275 y(asso)q(ciated)j (with)e(the)i(data)e(for)h(y)o(our)f(letter.)512 1347 y Fn(>)582 1351 y Fo(f\(F\);)450 1510 y Fe(Exercise)k(3)450 1591 y Fp(Example)11 b(1)h(from)f(Section)i(3.1)f(con)o(tains)g(a)g(matrix)f(M)h(whic)o(h)h (represen)o(ts)i(a)d("shear)450 1641 y(along)e(the)i(x-axis.")e(Call)g(this)h (shear)h(matrix)d Fj(S)1203 1647 y Fn(x)1236 1641 y Fp(and)i(apply)f(it)h(to) g(the)h(letter)g(stored)450 1690 y(in)h(F)h(b)o(y)g(forming)d(the)k(matrix)d (pro)q(duct)512 1742 y Fj(S)537 1748 y Fn(x)572 1742 y Fp(F.)i(Plot)f(the)i (image)d(of)h(the)i(sheared)g(letter.)450 1902 y Fe(Exercise)i(4)450 1982 y Fp(Construct)12 b(the)g(rotation)e(matrix)f(corresp)q(onding)j(to)f(a) f(coun)o(terclo)q(c)o(kwise)j(rotation)450 2032 y(of)f(20)f(degrees.)19 b(Apply)12 b(this)g(rotation)g(matrix)e(to)i(y)o(our)g(letter)h(and)f(plot)g (the)h(result.)512 2084 y(Since)h(y)o(ou)e(will)g(need)h(other)h(rotations,)e (it)h(will)e(b)q(e)j(more)e(e\016cien)o(t)h(if)f(y)o(ou)h(build)450 2134 y(R,)g(a)g("rotation)f(matrix)g(generating)i(function")f(that)g(accepts) j(as)d(input)g(the)i(n)o(um-)450 2184 y(b)q(er)21 b(of)e(degrees)j(\(coun)o (terclo)q(c)o(kwise\))g(through)e(whic)o(h)f(the)i(rotation)e(is)h(to)g(tak)o (e)450 2233 y(place,)13 b(and)h(returns)h(the)f(matrix)d(for)i(this)h (rotation.)j(Maple's)c(arro)o(w)h(notation)e(for)450 2283 y(building)g (functions)i(is)g(appropriate)g(here.)19 b(An)14 b(appropriate)g(syn)o(tax)g (w)o(ould)f(b)q(e)512 2354 y Fn(>)582 2358 y Fo(R)21 b(:=)h(x)f(->)h (matrix\(2,2,[cos\()o(x\),-s)o(in\(x\))o(,sin\()o(x\),c)o(os\(x\))o(]\);)512 2430 y Fn(>)582 2434 y Fo(R20)f(:=)g(R\(20*Pi/180\);)p eop bop 150 275 a Fm(3.1)13 b(Maple)h(On)g(Line)967 b(35)212 391 y Fp(Rotate)14 b(y)o(our)g(letter)h(b)o(y)f(m)o(ultiplyi)o(ng)d(it)j(b)o(y)g (R20)f(and)g(plot)h(the)g(rotated)h(letter)150 441 y(b)o(y)f(in)o(v)o(oking)e (the)i(plotting)f(function)h(f)f(built)g(in)h(Exercise)h(2.)150 590 y Fe(Exercise)i(5)150 668 y Fp(Create)11 b(another)g(letter,)h(reduce)g
(it)e(to)g(a)g(matrix)f(of)g(co)q(ordinates,)j(and)e(store)h(it)f(in)g(an)150 718 y(appropriate)k(v)n(ariable.)i(F)m(or)d(example,)f(the)i(letter)h(E)e (can)h(b)q(e)g(created)h(b)o(y)e(a)g(simple)150 768 y(mo)q(di\014cation)f(of) i(the)g(matrix)f(represen)o(ting)j(the)e(letter)i(F,)d(and)h(its)h(matrix)d (stored)150 817 y(in)g(the)i(v)n(ariable)d(E.)i(Plot)f(the)i(new)f(letter.)19 b(Then,)13 b(in)f(an)o(ticipation)f(of)i(plotting)e(the)150 867 y(com)o(bination)k(F)j(E,)g(determine)g(a)g(w)o(a)o(y)f(to)h(plot)f(the)h (result)h(of)e(shifting)h(E)g(three)150 917 y(units)c(to)g(the)g(righ)o(t.)k (Call)12 b(y)o(our)i(shifted)g(letter)h(TE.)212 968 y(Hin)o(t:)k(Since)14 b(the)h(translate)g(of)f(E)g(three)i(units)e(to)g(the)h(righ)o(t)f(w)o(ould)f (ha)o(v)o(e)h(eac)o(h)150 1017 y(x-co)q(ordinate)e(increased)h(b)o(y)e(3,)h (y)o(ou)f(w)o(an)o(t)g(to)h(\014nd)g(a)f(simple)f(w)o(a)o(y)h(to)h(add)f(3)h (to)f(eac)o(h)150 1067 y(elemen)o(t)16 b(in)h(the)g(\014rst)h(ro)o(w)e(of)g (the)i(matrix)d(E.)h(The)h(Maple)g(syn)o(tax)g(3$)o(7)g(writes)g(a)150 1117 y(sequence)i(of)d(sev)o(en)i(3's)e(separated)i(b)o(y)e(commas.)24 b(Hence,)18 b(the)g(follo)o(wing)c(matrix)150 1167 y(has)g(ro)o(w)g(of)f (threes)j(and)d(a)h(ro)o(w)g(of)f(zeros.)212 1234 y Fn(>)282 1238 y Fo(T)21 b(:=)h(matrix\(2,7,[3$7,)o(0$7]\))o(;)212 1318 y Fp(Next,)12 b(plot)f(the)i(com)o(bination)c(FE.)i(The)h(utilit)o(y)f (function)g(f)g(whic)o(h)h(w)o(e)f(built)g(for)150 1367 y(graphing)k(letters) h(is)f(not)h(sophisticated)g(enough)f(to)g(accept)i(m)o(ultiple)c(inputs)i (the)150 1417 y(w)o(a)o(y)e(the)i(standard)f(plot)g(functions)g(in)g(Maple)f (will.)18 b(Hence,)d(create)g(plots)f(of)f(eac)o(h)150 1467 y(letter)18 b(and)f(com)o(bine)f(the)i(resulting)g(graphics)f(ob)r(jects)i (with)e(the)h Fk(displa)o(y)c Fp(com-)150 1517 y(mand.)i(Assign)e(the)h(plot) e(of)g(eac)o(h)h(letter)h(to)f(a)f(v)n(ariable,)f(b)q(eing)i(sure)h(to)f (terminate)150 1567 y(eac)o(h)g(command)e(with)h(a)h(colon.)j(Then)e(in)o(v)o (ok)o(e)e Fk(displa)o(y)p Fp(.)212 1634 y Fn(>)282 1638 y Fo(p1:=f\(TE\):)212 1705 y Fn(>)282 1709 y Fo(p2:=f\(F\):)212 1777 y Fn(>)282 1781 y Fo(display\([p1,p2]\))o(;)150 1930 y Fe(Exercise)k(6)150 2008 y Fp(Let)c(S)f(b)q(e)h(the)g(transformation)e(of)g Fj(R)742 1992 y Fi(2)773 2008 y Fp(to)h(itself)g(wherein)h(S\()p Fk(X)p Fp(\))g(is)f(a)g(shift)h(of)e Fk(X)i Fp(one)150 2057 y(unit)j(to)h(the)g (righ)o(t.)26 b(Sho)o(w)16 b(graphically)f(that)i(S)f(is)h(not)f(linear.)26 b(Sp)q(eci\014cally)m(,)16 b(use)150 2107 y(the)e(letter)h(created)h(in)d (Exercise)j(1)d(to)h(sho)o(w)g(that)g(S\(2)g Fk(X)p Fp(\))g Fg(6)p Fp(=)21 b(2)13 b(S\()p Fk(X)p Fp(\).)150 2256 y Fe(Exercise)k(7)150 2334 y Fp(F)m(or)e(eac)o(h)h(of)e(the)i(follo)o(wing,)c(\014nd)k(a)f(matrix)e (M)i(for)g(whic)o(h)g(m)o(ultiplication)d(w)o(ould)150 2384 y(accomplish)d(the)i(indicated)f(transformation.)15 b(In)10 b(eac)o(h)h(case,)h(v)n(alidate)d(y)o(our)h(matrix)150 2434 y(b)o(y)k(applying)e(it)i(to)f(the)i(letter)g(created)g(in)f(Exercise)h(1.)p eop bop 450 275 a Fp(36)967 b Fm(3.2)13 b(Maple)h(On)g(Line)450 391 y Fe(P)n(art)19 b(\(a\))450 473 y Fp(Ma)14 b(\015ips)g(a)f(letter)i (upside)f(do)o(wn.)450 603 y Fe(P)n(art)19 b(\(b\))450 685 y Fp(Mb)14 b(\015ips)g(a)f(letter)i(left-to-righ)o(t.)450 815 y Fe(P)n(art)k(\(c\))450 896 y Fp(Mc)14 b(rotates)h(a)f(letter)g(b)o(y)g(20)f (degrees)j(coun)o(terclo)q(c)o(kwise)450 1027 y Fe(P)n(art)j(\(d\))450 1108 y Fp(Md)14 b(is)g(a)f(shear)i(along)e(y-axis.)450 1273 y Fe(Exercise)k(8)450 1355 y Fp(Plot)12 b(the)h(e\013ect)h(of)d(eac)o(h)i(of) e(the)i(follo)o(wing)d(transformations)h(applied)g(to)h(the)h(letter)450 1405 y(created)i(in)f(Exercise)h(1.)450 1535 y Fe(P)n(art)k(\(a\))450 1616 y Fp(A)10 b(shear)g(along)f(the)h(x-axis)f(follo)o(w)o(ed)g(b)o(y)g (rotation)g(of)g(20)h(degrees)h(coun)o(terclo)q(c)o(kwise.)450 1781 y Fe(P)n(art)19 b(\(b\))450 1863 y Fp(Rotate)14 b(20)f(degrees)j(coun)o (terclo)q(c)o(kwise,)e(then)h(shear)g(along)d(the)j(x-axis.)450 2028 y Fe(P)n(art)k(\(c\))450 2110 y Fp(Shear)14 b(along)f(the)i(x-axis,)d (follo)o(w)o(ed)h(b)o(y)g(shear)i(along)e(the)h(y-axis.)450
2275 y Fe(P)n(art)19 b(\(d\))450 2356 y Fp(Shear)14 b(along)f(the)i(y-axis,)d (follo)o(w)o(ed)h(b)o(y)g(a)h(shear)h(along)d(the)j(x-axis.)p eop bop 150 275 a Fm(3.2)13 b(Maple)h(On)g(Line)967 b(37)150 391 y Fq(3.2)89 b(On)22 b(Line)150 482 y Fp(Restart)e(Maple)g(to)f(clear)h(its)f (memory)e(of)i(all)g(v)n(ariables,)g(then)i(re-initialize)d(b)o(y)150 532 y(loading)12 b(b)q(oth)i(the)h Fl(linalg)e Fp(and)h Fl(plots)g Fp(pac)o(k)n(ages.)212 582 y(These)f(exercises)g(will)d(con)o(tin)o(ue)i(the) g(study)g(the)g(geometric)e(asp)q(ect)j(of)e(transfor-)150 632 y(mations)g(in)h Fj(R)385 617 y Fi(3)404 632 y Fp(.)17 b(F)m(or)c(this)f(w)o(ork)h(it)f(will)f(b)q(e)j(useful)f(to)f(again)g (de\014ne)i(the)f(function)f(f)150 681 y(used)j(in)f(On)g(Line)g(Section)h (3.1.)i(In)d(that)g(section)h(the)g(function)f(to)q(ok)f(in)h(a)g(matrix)150 731 y(represen)o(ting)i(a)d(plane)h(\014gure,)g(and)f(returned)j(a)e(plot)f (of)g(the)i(\014gure.)150 874 y Fe(Exercise)i(1)150 950 y Fp(In)h(place)f(of) g(the)h(letters)h(of)e(the)h(alphab)q(et)g(that)g(w)o(ere)g(used)h(in)e(the)h (exercises)i(of)150 1000 y(Section)g(3.1,)f(use)g(line)g(segmen)o(ts)g(to)g (create)i(the)f(outline)e(of)h(a)g(car.)34 b(Dra)o(w)18 b(the)150 1050 y(outline)12 b(on)h(a)g(piece)g(of)g(pap)q(er,)g(ev)o(en)g(a)g(sheet)h (of)f(graph)f(pap)q(er,)i(pic)o(k)e(for)h(the)g(origin)150 1100 y(the)j(endp)q(oin)o(t)f(of)f(one)h(segmen)o(t,)f(and)h(\014nd)g(the)h (co)q(ordinates)f(of)g(the)g(endp)q(oin)o(ts)h(of)150 1150 y(all)f(the)i(line)f(segmen)o(ts.)26 b(En)o(ter)18 b(these)g(co)q(ordinates)f (as)f(columns)g(of)g(a)g(matrix)e(C.)150 1199 y(Hence,)h(y)o(ou)e(migh)o(t)f (ha)o(v)o(e)i(a)f(matrix)f(suc)o(h)j(as)f(the)h(follo)o(wing.)212 1264 y Fn(>)282 1268 y Fo(C)21 b(:=)h(matrix\([[0,1,3/2)o(,5,6,)o(0],[0)o (,1,5/)o(2,2,0)o(,0]])o(\);)212 1345 y Fp(represen)o(ting)16 b(the)e(car)h(whose)f(shap)q(e)h(is)e(giv)o(en)h(b)o(y)g(the)g(follo)o(wing)d (graph.)212 1410 y Fn(>)282 1414 y Fo(f\(C\);)212 1491 y Fp(F)m(or)21 b(the)g(reader)h(who)e(b)q(eliev)o(es)i(this)f(car)g(lo)q(oks)f(more)g(lik)o (e)g(a)g(\015at-iron,)i(w)o(e)150 1541 y(giv)o(e)e(the)g(data)g(p)q(oin)o(ts) g(for)g(a)g(car)g(that)g(is)g(signi\014can)o(tly)g(b)q(etter)h(lo)q(oking.)36 b(The)150 1591 y(in)o(terested)19 b(reader)g(can)f(en)o(ter)h(the)f(data)f (in)o(to)g(an)h(appropriate)f(matrix)f(and)i(run)150 1640 y(the)c(exp)q (erimen)o(ts)h(in)e(these)i(exercises)h(with)e(the)g(impro)o(v)o(ed)f(image.) 212 1690 y([[0,)e(.954e-1],)f([.104e-1,)g(.1612],)g([.1425,)f(.2237],)h ([.2124,)g(.2336],)g([.2513,)f(.2401],)150 1740 y([.2902,)17 b(.2401],)g([.3187,)g(.3158],)g([.3472,)g(.3553],)g([.3912,)g(.3684],)f ([.4611,)h(.3750],)150 1790 y([.5104,)g(.3783],)g([.5674,)g(.3783],)g ([.6373,)g(.3783],)g([.6710,)g(.3684],)f([.6891,)h(.3520],)150 1840 y([.7047,)g(.3388],)g([.7202,)g(.3191],)g([.7306,)g(.2961],)g([.7409,)g (.2763],)f([.7876,)h(.2664],)150 1889 y([.8135,)g(.2599],)g([.8264,)g (.2500],)g([.8394,)g(.2368],)g([.8472,)g(.2072],)f([.8497,)h(.1612],)150 1939 y([.8497,)11 b(.1513],)h([.8497,)f(.1283],)h([.8497,)f(.1020],)h ([.26e-2,)g(.1053]])212 1989 y(T)m(ransform)h(y)o(our)i(plane)g(image)e(in)o (to)h(a)h(3d)g(ob)r(ject)g(b)o(y)g(altering)f(the)i(matrix)d(C)150 2039 y(as)h(follo)o(ws.)k(Add)d(a)f(middle)e(ro)o(w)i(of)g(zeros)h(b)o(y)f (\014rst)i(adding)d(a)h(third)g(ro)o(w)g(of)g(zeros,)150 2089 y(then)i(sw)o(apping)f(the)i(second)g(and)e(third)h(ro)o(ws.)24 b(This)15 b(can)h(b)q(e)g(done)g(in)g(Maple)f(b)o(y)150 2138 y(\014rst)f(stac)o(king)f(C)h(with)f(a)g(ro)o(w)g(of)g(zeros,)h(then)g(using) g(the)g Fk(sw)o(apro)o(w)e Fp(command)f(to)150 2188 y(in)o(terc)o(hange)k (the)f(new)g(ro)o(w)g(of)f(zeros)i(with)f(the)g(original)e(ro)o(w)i(2.)212 2253 y Fn(>)282 2257 y Fo(d)21 b(:=)h(swaprow\(stack\(C,)o([0$6])o(\),2,3)o (\);)212 2334 y Fp(Since)f(sev)o(eral)h(3d)e(plots)g(will)g(b)q(e)h (required,)i(it)d(is)g(v)o(ery)h(useful)g(to)g(de\014ne)g(a)150 2384 y(function)14 b(f3)g(that)g(will)f(tak)o(e)h(in)g(a)g(matrix)f(represen) o(ting)j(a)e(3d)g(ob)r(ject,)h(and)f(return)150 2434 y(a)g(3d)f(plot)h(of)f (the)h(ob)r(ject)h(so)f(represen)o(ted.)p eop
bop 450 275 a Fp(38)967 b Fm(3.2)13 b(Maple)h(On)g(Line)512 387 y Fn(>)582 391 y Fo(f3)21 b(:=)h(u->)f(pointplot3d\([seq)o(\(conv)o (ert\(c)o(ol\(u,)o(k\),l)o(ist\),)o(k=1..)o(coldi)o(m\(u\)\))o(],)512 434 y(style=line,)f(axes=boxed,)f(scaling=constrained)o(,)g(color=black,)g (labels=[x,z,y],)512 476 y(labelfont=[TIMES,BO)o(LD,14)o(]\):)512 554 y Fp(Ha)o(ving)10 b(de\014ned)h(the)g(function)g(f3,)f(apply)f(it)h(to)h (the)g(matrix)d(d)j(whic)o(h)f(represen)o(ts)450 604 y(a)h(\014rst)i(v)o (ersion)f(of)f(a)h(3d)f(car.)18 b(Since)12 b(this)g(is)g(a)f(3d)h(plot,)f(it) g(can)h(b)q(e)h(rotated)f(in)f(Maple)450 654 y(b)o(y)i(clic)o(king)g(on)g (the)h(image)d(and)i(then)h(using)f(the)h(mouse)f(to)g("grab")g(and)g ("rotate")450 704 y(the)h(b)q(ounding)g(b)q(o)o(x.)k(Clic)o(king)12 b(on)i(the)g(R)g(in)f(the)h(to)q(olbar)g(redra)o(ws)h(the)f(graph.)512 770 y Fn(>)582 774 y Fo(f3\(d\);)450 921 y Fe(Exercise)j(2)450 999 y Fp(F)m(urther)f(dimension)d(can)h(b)q(e)i(added)f(to)f(the)i(image)c (of)i(the)i(car)f(b)o(y)f(adding)g(1/4)g(to)450 1048 y(eac)o(h)g(elemen)o(t)f (in)g(ro)o(w)g(3)h(of)e(the)j(matrix)c(d,)i(then)i(augmen)o(ting)c(the)j (matrix)e(d)h(with)450 1098 y(the)g(altered)f(v)o(ersion)h(e.)18 b(One)12 b(w)o(a)o(y)g(to)g(do)g(this)g(is)g(to)g(assem)o(ble)f(\(via)h(stac) o(k\))g(the)h(\014rst)450 1148 y(ro)o(w)h(of)f(d,)g(the)i(altered)f(second)h (ro)o(w,)f(and)f(then)i(the)f(third)g(ro)o(w.)512 1214 y Fn(>)582 1218 y Fo(e)21 b(:=)h(stack\(row\(d,1\),e)o(valm\()o(row\(d)o(,2\)+1)o (/4\),r)o(ow\(d)o(,3\)\);)512 1285 y Fn(>)582 1289 y Fo(F)f(:=)h (augment\(d,e\);)512 1367 y Fp(T)m(est)15 b(the)g(e\016cacy)g(of)f(these)h (impro)o(v)o(emen)o(ts)d(b)o(y)i(plotting,)f(using)h(the)h(function)450 1417 y(f3.)450 1564 y Fe(Exercise)i(3)450 1641 y Fp(Add)g(some)f(substance)j (to)e(the)h(car)f(b)o(y)g(sk)o(etc)o(hing)g(in)g(diagonal)e(lines)i(on)g(eac) o(h)h(of)450 1691 y(the)d(narro)o(w)g(faces.)21 b(This)15 b(requires)h (alternating)e(the)h(columns)e(of)i(the)g(augmen)o(ted)450 1741 y(matrix)d(F)h(so)g(that)h(the)g(\014rst)g(column)e(comes)h(from)e(d,)i (the)h(second)h(from)c(matrix)h(e,)450 1791 y(etc.)27 b(In)16 b(e\013ect,)i(build)e(a)g(new)h(matrix)d(FF)j(b)o(y)f(augmen)o(ting)e(pairs)j (of)e(columns)h(of)450 1841 y(the)e(form)f([)g Fj(d)667 1847 y Fn(k)694 1841 y Fj(e)713 1847 y Fn(k)734 1841 y Fp(].)k(This)d(is)g (accomplished)f(in)g(Maple)h(via)f(the)h(syn)o(tax)512 1907 y Fn(>)582 1911 y Fo(FF)21 b(:=)h(augment\(F,seq\(o)o(p\([co)o(l\(d,k)o (\),col)o(\(e,k\))o(]\),k)o(=1..c)o(oldim)o(\(C\)\)\))o(:)512 1990 y Fp(The)15 b(v)n(alidation)c(of)i(the)i(manipulation)c(is)i(in)h(the)g (plotting.)512 2056 y Fn(>)582 2060 y Fo(f3\(FF\);)450 2207 y Fe(Exercise)j(4)450 2284 y Fp(Some)e(of)h(the)h(transformations)d(that)j (will)e(b)q(e)h(applied)g(to)g(the)h(car)g(include)f(rota-)450 2334 y(tions.)i(T)m(o)c(k)o(eep)g(the)h(rotated)g(car)f(in)g(the)g(viewing)g (windo)o(w,)e(it)i(will)f(help)h(to)g(mo)o(v)o(e)450 2384 y(the)i(origin)e (to)h(the)h(cen)o(ter)h(of)e(the)h(car.)23 b(Deduce)17 b(the)f(co)q (ordinates)g(of)e(this)i(cen)o(ter,)450 2434 y(and)j(mo)o(v)o(e)f(the)h (origin)f(to)h(that)h(p)q(oin)o(t)e(b)o(y)h(subtracting)h(appropriate)f (constan)o(ts)p eop bop 150 275 a Fm(3.3)13 b(Maple)h(On)g(Line)967 b(39)150 391 y Fp(from)15 b(the)j(\014rst)f(and)g(third)g(ro)o(ws)g(of)f(the)i(matrix)d (FF.)h(Remem)o(b)q(er)g(that)h(the)g(\014rst)150 441 y(ro)o(w)10 b(records)i(x-co)q(ordinates,)e(the)h(second)g(ro)o(w,)f(z-co)q(ordinates,)i (and)e(the)h(third)f(ro)o(w,)150 491 y(y-co)q(ordinates.)18 b(F)m(orm)12 b(this)i(new)g(matrix)e(b)o(y)i(altering)f(the)h(appropriate)g (ro)o(ws)g(and)150 541 y(reassem)o(bling)f(them)g(in)o(to)g(a)h(new)g (matrix.)212 606 y Fn(>)282 610 y Fo(FFF)21 b(:=)g(stack\(evalm\(row\(FF,)o (1\)-3\))o(,row\()o(FF,2\))o(,eva)o(lm\(ro)o(w\(FF,)o(3\)-3/)o(2\)\):)212 688 y Fp(Next,)f(rotate)e(the)h(\014gure)g(coun)o(terclo)q(c)o(kwise)h(30)d (degrees)j(ab)q(out)e(the)h(x-axis,)150 738 y(then)c(rotate)f(that)g(image)e (20)i(degrees)i(coun)o(terclo)q(c)o(kwise)f(ab)q(out)f(the)h(z-axis.)j(This) 150 787 y(is)c(most)f(easily)h(done)g(b)o(y)g(building)f(functions)h(that)g
(yield)g(the)h(appropriate)f(three-)150 837 y(dimensional)j(rotation)h (matrices)h(\(Exercises)i(10,)f(11,)f(and)g(12)g(in)g(Section)g(3.1\),)150 887 y(then)c(in)o(v)o(oking)d(the)j(functions)g(for)f(the)h(required)g (angles.)k(\(See)d(Exercise)f(4)f(in)g(the)150 937 y(On)g(Line)g(section)h (for)e(Section)h(3.1.\))150 1081 y Fe(Exercise)j(5)150 1158 y Fp(Obtain)g(a)f(single)h(matrix)e(whose)i(action)f(under)i(m)o (ultiplicatio)o(n)c(repro)q(duces)19 b(the)150 1208 y(t)o(w)o(o)e(successiv)o (e)i(rotations)e(implemen)o(ted)e(in)h(Exercise)j(4.)27 b(V)m(alidate)16 b(y)o(our)h(single)150 1258 y(matrix)12 b(b)o(y)i(again)e(plotting)h(the)i (rotated)f(car.)150 1403 y Fe(Exercise)j(6)150 1480 y Fp(What)c(image)e(w)o (ould)h(y)o(ou)g(see)j(if)d(y)o(ou)g(transformed)h(the)g(matrix)f(FFF)h(b)o (y)g(a)f(rank)h(2)150 1529 y(transformation?)j(Create)c(a)g(random)e(rank)i (2)g(matrix)e(and)h(test)i(y)o(our)f(guess.)18 b(After)150 1579 y(prin)o(ting)c(graphs)g(of)g(the)h(transformed)f(and)g(un)o (transformed)g(images,)e(attempt)i(to)150 1629 y(lab)q(el)g(sev)o(eral)i(p)q (oin)o(ts)f(where)h(the)f(transformation)e(is)i(man)o(y-to-one.)k(The)d(en)o (tries)150 1679 y(in)11 b(the)g(transformation)f(matrix)f(should)i(b)q(e)g (random)f(n)o(um)o(b)q(ers)h(in)f(the)i(range)g([-1,1],)150 1729 y(lest)i(the)h(scale)f(of)f(the)i(car)f(b)q(e)h(altered)f(completely)m (.)150 1873 y Fe(Exercise)j(7)150 1950 y Fp(What)e(image)f(w)o(ould)h(y)o(ou) g(see)i(if)d(y)o(ou)i(transformed)e(the)j(matrix)c(FFF)j(b)o(y)g(a)f(rank)150 2000 y(1)j(transformation?)30 b(Create)19 b(a)g(random)d(rank)j(1)f(matrix)e (and)j(test)g(y)o(our)f(guess.)150 2050 y(Again,)d(b)q(e)h(sure)h(to)e (restrict)i(the)f(en)o(tries)h(in)e(y)o(our)g(random)f(matrix)g(to)i(the)g (range)150 2100 y([-1,1].)150 2293 y Fq(3.3)66 b(On)22 b(Line)150 2384 y Fp(Restart)f(Maple)g(to)g(clear)g(its)g(memory)d(of)i(all)g(v)n (ariables,)h(then)g(reinitialize)f(b)o(y)150 2434 y(loading)12 b(the)j Fl(linalg)p Fp(,)e Fl(plots,)h Fp(and)g Fl(plotto)n(ols)f Fp(pac)o(k)n(ages.)p eop bop 450 275 a Fp(40)967 b Fm(3.3)13 b(Maple)h(On)g(Line)450 391 y Fe(Exercise)j(1)450 469 y Fp(Create)f(M,)e(a)h(random)e(2)i(x)f(3)h (matrix)e(with)i(rank)f(1.)21 b(\(See)16 b(Exercise)h(3)d(in)h(Section)450 519 y(1.4.\))24 b(If)16 b(M)g(is)h(in)o(terpreted)g(as)g(the)g(matrix)d(of)i (a)g(transformation)e(acting)i(on)g Fj(R)1770 503 y Fi(3)1788 519 y Fp(,)450 568 y(what)e(should)h(the)g(dimension)e(of)g(the)j(image)c(of) i(this)g(transformation)f(b)q(e?)20 b(V)m(erify)450 618 y(this)12 b(b)o(y)f(creating)h(100)f(random)f(p)q(oin)o(ts)i(in)f Fj(R)1166 603 y Fi(3)1185 618 y Fp(,)g(transforming)f(them)h(under)i(M,)e(and)450 668 y(plotting)i(the)h(p)q(oin)o(ts.)512 718 y(Next,)21 b(generate)g(P)m(,)e (a)g(matrix)e(con)o(taining)i(100)f(random)g(p)q(oin)o(ts)i(in)f Fj(R)1704 703 y Fi(3)1722 718 y Fp(.)35 b(A)450 768 y(momen)o(t's)14 b(though)o(t)j(ab)q(out)g(ho)o(w)f(the)i(transformed)e(p)q(oin)o(ts)h(will)e (b)q(e)i(plotted)g(will)450 818 y(determine)d(the)h(optim)o(um)c(strategy)k (for)e(generating)i(and)f(plotting)f(the)i(p)q(oin)o(ts.)j(If)450 868 y(the)13 b(p)q(oin)o(ts)f(are)h(stored)g(as)f(columns)f(of)h(a)g(matrix,) e(they)j(can)g(b)q(e)f(plotted)h(b)o(y)f(apply-)450 917 y(ing)j(the)g Fk(p)q(oin)o(tplot)c Fp(command)i(to)i(the)h(matrix.)k(Hence,)c(let)g(P)f(b)q (e)h(of)f(dimension)450 967 y(3)f(x)g(100)g(so)g(the)h(pro)q(duct)h(M)e(P)h (will)d(b)q(e)j(2)g(x)f(100.)k(Create)e(P)e(b)o(y)h(juxtap)q(osing,)e(via)450 1017 y Fk(augmen)o(t)p Fp(,)i(100)h(v)o(ectors)i(generated)h(b)o(y)d Fk(randv)o(ector)p Fp(.)25 b(T)m(erminating)15 b(the)i(com-)450 1067 y(mands)12 b(with)h(a)h(colon)f(\(:\))18 b(signals)12 b(Maple)i(not)f(to)g(prin)o(t)h(the)g(rather)g(large)f(outputs)450 1117 y(to)k(the)g(screen.)28 b(Finally)m(,)15 b(an)o(ticipating)g(Exercise)k (2)d(where)i(this)f(plot)f(will)f(b)q(e)i(re-)450 1167 y(quired,)g(store)g (the)g(plot)f(data)f(structure)k(in)d(a)g(v)n(ariable,)f(sa)o(y)h(p1,)g(so)h (later,)f(other)450 1216 y(images)c(can)i(b)q(e)h(sup)q(erimp)q(osed)f(on)g (it.)512 1283 y Fn(>)582 1287 y Fo(P)21 b(:=)h(augment\(seq\(rand)o(vecto)o (r\(3\),)o(k=1..)o(100\)\))o(:)512 1353 y Fn(>)582 1357 y Fo(S)f(:=)h
(evalm\(M&*P\):)512 1424 y Fn(>)582 1428 y Fo(p1)f(:=)h(pointplot\(S\):)512 1494 y Fn(>)582 1498 y Fo(p1;)450 1617 y Fe(Exercise)17 b(2)450 1694 y Fp(The)i(plot)g(generated)h(in)e(Exercise)j(1)d(should)h(sho)o(w)g (the)h(span)f(of)f(an)o(y)g(non-zero)450 1744 y(column)c(of)i(M.)f (Demonstrate)h(this)g(b)o(y)g(c)o(ho)q(osing)g(a)g(column)e(of)i(M)g(and)f (plotting,)450 1794 y(on)10 b(the)h(graph)g(from)e(Exercise)j(1,)e(100)g (random)f(p)q(oin)o(ts)h(in)h(the)g(span)f(of)g(this)h(column.)512 1844 y(De\014ne)19 b(the)f(function)g(f)g(whic)o(h)f(generates)j(a)e(random)e (in)o(teger)i(in)g(the)g(closed)450 1894 y(in)o(terv)n(al)f([-500,500].)29 b(Then,)19 b(using)f(the)h Fk(seq)f Fp(command,)e(giv)o(e)i(to)g Fk(p)q(oin)o(tplo)o(t)d Fp(a)450 1943 y(sequence)f(of)c(100)h(random)f(m)o (ultiples)f(of)i(the)h(\014rst)g(column)e(of)h(M.)f(Color)h(the)h(p)q(oin)o (ts)450 1993 y(red)j(and)f(assign)f(the)i(plot)e(data)h(structure)i(to)e(a)g (v)n(ariable,)e(sa)o(y)i(p2.)k(After)d(viewing)450 2043 y(the)j(graph,)f (merge)f(it)h(with)f(the)i(plot)e(from)g(Exercise)i(1)f(b)o(y)g(use)h(of)e (the)i Fk(displa)o(y)450 2093 y Fp(command.)27 b(Assign)17 b(this)h(comp)q(osite)f(graph)g(to)h(a)f(v)n(ariable,)g(sa)o(y)h(p3,)g(for)f (use)h(in)450 2143 y(Exercise)e(3.)512 2209 y Fn(>)582 2213 y Fo(f)21 b(:=)h(rand\(-500..500\):)512 2280 y Fn(>)582 2284 y Fo(p2)f(:=)h(pointplot\([seq\()o(evalm)o(\(col\()o(M,1\)*)o(f\(\)\),)o (k=1.)o(.100\))o(],)d(color)512 2326 y(=)j(red\):)42 b(p2;)512 2392 y Fn(>)582 2396 y Fo(p3)21 b(:=)h(display\([p1,p2])o(\):)41 b(p3;)p eop bop 150 275 a Fm(3.4)13 b(Maple)h(On)g(Line)967 b(41)150 391 y Fe(Exercise)17 b(3)150 481 y Fp(Using)e(information)e(and)j(insigh)o(ts)f (from)f(Exercises)k(1)d(and)g(2,)h(\014nd)f(\(a\))h(a)f(sp)q(eci\014c)150 531 y(v)o(ector)f Fk(B)g Fp(in)f Fj(R)402 516 y Fi(2)434 531 y Fp(for)g(whic)o(h)g(the)h(equation)f(M)h Fk(X)g Fp(=)f Fk(B)h Fp(is)f Fl(not)h Fp(solv)n(able;)e(and)h(\(b\))h(a)150 581 y(v)o(ector)f Fk(C)g Fp(in)e Fj(R)398 566 y Fi(2)429 581 y Fp(for)h(whic)o(h)g(the)h(equation)f(M)g Fk(X)g Fp(=)h Fk(C)g Fl(is)e Fp(solv)n(able.)17 b(Indicate)c(these)150 630 y(v)o(ectors)f(on)e (the)h(comp)q(osite)f(graph)h(pro)q(duced)g(in)g(Exercise)h(2.)k(V)m(erify)11 b(y)o(our)f(answ)o(ers)150 680 y(b)o(y)16 b(computing,)e(via)h Fk(rref)p Fp(,)g(the)h(reduced)i(ro)o(w)e(ec)o(helon)g(forms)f(of)g(the)i (augmen)o(ted)150 730 y(matrices)c([M,)p Fk(B)p Fp(])g(and)h([M,)p Fk(C)p Fp(].)150 932 y Fe(Exercise)j(4)150 1021 y Fp(Plot)j(100)f(random)g (elemen)o(ts)h(from)e(the)j(n)o(ull)e(space)j(of)d(M.)h(Maple's)g(n)o (ullspace)150 1071 y(command)9 b(will)h(pro)o(vide)h(a)g(basis)h(for)f(the)h (n)o(ull)f(space)h(of)f(M.)g(This)g(basis)h(is)f(returned)150 1121 y(as)17 b(a)g(set)h(of)e(v)o(ectors)i(whic)o(h)f(can)g(b)q(e)h (extracted)g(from)e(the)h(set)h(with)f(the)g(brac)o(k)o(et)150 1171 y(notation.)29 b(The)18 b Fk(seq)f Fp(command)e(can)j(b)q(e)g(used)h(to) e(generate)i(a)f(sequence)i(of)d(100)150 1221 y(random)f(linear)g(com)o (binations)f(of)i(these)i(basis)e(v)o(ectors,)h(a)f(sequence)j(whic)o(h)d (can)150 1270 y(then)f(b)q(e)g(plotted)f(in)g Fj(R)530 1255 y Fi(3)564 1270 y Fp(with)f(the)i Fk(p)q(oin)o(tplot)o(3d)c Fp(command.)19 b(The)d(3d)f(graph)g(so)150 1320 y(generated)f(can)f(b)q(e)g (rotated)h(on-screen)g(b)o(y)f(grabbing)e(and)i(rotating)f(the)h(b)q(ounding) 150 1370 y(b)q(o)o(x.)k(It)c(should)f(then)h(b)q(e)g(p)q(ossible)g(to)f (observ)o(e)h(the)g(nature)g(of)f(that)h(p)q(ortion)f(of)f(the)150 1420 y(span)j(so)g(generated.)212 1477 y(Finally)m(,)e(explain)h(ho)o(w)g (this)h(plot)f(relates)i(to)f(the)h(Rank-Nullit)o(y)c(Theorem.)150 1678 y Fe(Exercise)17 b(5)150 1768 y Fp(Randomly)8 b(generate)k(a)e(rank)g(2) h(matrix)d(M)j(of)f(dimension)f(3)h(x)g(3.)17 b(Rep)q(eat)11 b(Exercises)150 1818 y(1)j(through)g(4,)f(suitably)g(mo)q(di\014ed)f(to)i (accoun)o(t)g(for)g(the)g(di\013eren)o(t)h(dimensions.)212 1874 y(Sp)q(eci\014cally)m(,)j(this)g(means)f(y)o(ou)g(are)h(to)g(generate)h (M.)f(Then,)g(in)g(imitation)d(of)150 1924 y(Exercise)h(1,)d(the)i(matrix)e
(P)h(con)o(taining)f(100)h(random)f(p)q(oin)o(t)g(in)h Fj(R)1231 1909 y Fi(3)1250 1924 y Fp(,)g(and)g(plot)f(the)150 1974 y(pro)q(duct)j(M)e (P)m(.)g(In)h(imitation)c(of)j(Exercise)j(2,)d(plot)g(100)g(random)f(linear)h (com)o(bina-)150 2024 y(tions)i(of)f(the)i(columns)d(of)i(M.)f(In)h (imitation)d(of)j(Exercise)h(3,)f(\014nd)g(v)o(ectors)h Fk(B)f Fp(and)150 2074 y Fk(C)g Fp(for)g(whic)o(h)f(the)i(systems)e(M)h Fk(X)g Fp(=)g Fk(B)p Fp(,)f(and)h(M)g Fk(X)f Fp(=)h Fk(C)h Fp(are)f(not)f(solv)n(able,)g(and)150 2123 y(solv)n(able,)c(resp)q(ectiv)o (ely)m(.)19 b(V)m(erify)12 b(y)o(our)h(c)o(hoices)g(of)f Fk(B)h Fp(and)f Fk(C)h Fp(computationally)m(.)i(Fi-)150 2173 y(nally)m(,)f(in)h (imitation)e(of)i(Exercise)i(4,)e(plot)g(100)g(randomly)f(c)o(hosen)i(elemen) o(ts)g(from)150 2223 y(the)e(n)o(ull)f(space)i(of)f(M.)212 2280 y(When)j(y)o(ou)f(are)h(done,)g(don't)f(forget)h(to)f(relate)h(y)o(our)g (\014ndings)f(to)h(the)g(Rank-)150 2330 y(Nullit)o(y)c(Theorem.)p eop bop 450 275 a Fp(42)967 b Fm(3.4)13 b(Maple)h(On)g(Line)450 391 y Fq(3.4)66 b(On)22 b(Line)450 503 y Fp(Restart)f(Maple)g(to)g(clear)g (its)g(memory)d(of)i(all)g(v)n(ariables,)h(and)f(re-initialize)g(b)o(y)450 553 y(loading)12 b(the)j Fl(linalg)e Fp(and)h Fl(student)g Fp(pac)o(k)n(ages.)512 614 y(Maple)20 b(con)o(tains)f(a)h(v)n(ariet)o(y)f(of) g("solv)o(ers")h(for)f(equations)h(of)f(v)n(arious)g(t)o(yp)q(es.)450 663 y(F)m(or)g(example,)g(the)h(standard)g(sym)o(b)q(olic)d(solv)o(er)j(for)f (one)g(or)h(sev)o(eral)f(equations,)450 713 y(linear)13 b(and)h(non-linear)f (alik)o(e,)f(is)h Fk(solv)o(e)p Fp(.)k(The)d(standard)g(n)o(umeric)f(solv)o (er)h(for)f(suc)o(h)450 763 y(equations)g(w)o(ould)f(b)q(e)h Fk(fsolv)o(e)e Fp(\(\015oating)h(p)q(oin)o(t)g(solv)o(e\).)18 b(Di\013eren)o(tial)12 b(equations)h(are)450 813 y(solv)o(ed)d(b)o(y)g Fk(dsolv)o(e)p Fp(,)e(di\013erence)k(equations)e(are)g(solv)o(ed)g(b)o(y)g Fk(rsolv)o(e)e Fp(\(recursiv)o(e)k(solv)o(e\))450 863 y(and)i(Diophan)o(tine) f(equations)g(are)i(solv)o(e)e(b)o(y)h Fk(isolv)o(e)e Fp(\(in)o(teger)i(solv) o(e\).)512 923 y(In)20 b(the)g(linalg)e(pac)o(k)n(age,)j(if)e(a)g(set)i(of)e (linear)h(equations)f(is)h(captured)h(in)e(the)450 973 y(matrix-v)o(ector)d (format)f(A)i Fk(X)g Fp(=)g Fk(B)p Fp(,)g(then)g Fk(linsolv)o(e)d Fp(can)j(b)q(e)h(used.)28 b(Both)17 b Fk(solv)o(e)450 1023 y Fp(and)g Fk(linsolv)o(e)c Fp(return)18 b(general)f(sym)o(b)q(olic)f (solutions)g(when)h(applicable.)27 b(In)16 b(fact,)450 1073 y Fk(linsolv)o(e)9 b Fp(will)h(ev)o(en)j(return)g(solutions)f(in)f(terms)h (of)f(arbitrary)h(parameters.)18 b(Other)450 1123 y(approac)o(hes)d(to)g(the) g(solution)f(of)g(linear)g(systems)g(include)h(the)g(use)h(of)e Fk(gausselim)450 1173 y Fp(\(follo)o(w)o(ed)f(b)o(y)h Fk(bac)o(ksub)p Fp(\))g(or)g Fk(rref)f Fp(\(also)h(follo)o(w)o(ed)f(b)o(y)h Fk(bac)o(ksub)p Fp(\).)19 b(Maple's)14 b(com-)450 1222 y(mand)d(structure)k (is)d(ric)o(h)h(enough)f(that)h(nearly)f(an)o(y)h(undergraduate)g (mathematics)450 1272 y(that)f(can)f(b)q(e)h(articulated)g(in)f(standard)h (mathematical)c(notation)j(can)g(probably)g(b)q(e)450 1322 y(implemen)o(ted)h(in)h(the)i(con)o(text)f(of)f(Maple's)h(built-in)f (commands.)512 1383 y(There)g(are)g(t)o(w)o(o)f(cautions)g(to)g(observ)o(e)h (when)f(using)g(Maple)g(to)g(solv)o(e)g(linear)g(sys-)450 1433 y(tems.)24 b(First,)17 b(if)e(the)i(calculation)e(is)h(done)h(in)e (\015oating)g(p)q(oin)o(t)h(arithmetic,)f(Maple)450 1482 y(is)f(as)g(liable)e (to)i(round-o\013)g(and)g(truncation)g(errors)h(as)f(an)o(y)g(other)g(n)o (umeric)f(utilit)o(y)m(.)450 1532 y(Second,)h(when)g(w)o(orking)f(sym)o(b)q (olically)l(,)e(exact)j(expressions)h(for)e(n)o(um)o(b)q(ers)h(can)g(get)450 1582 y(daun)o(tingly)d(large,)g(thereb)o(y)j(consuming)c(memory)g(and)i (time.)k(Hence,)d(Maple)f(can-)450 1632 y(not)18 b(solv)o(e)f(sym)o(b)q (olically)e(systems)j(as)g(large)f(as)h(some)f(strictly)h(n)o(umeric)f (utilities)450 1682 y(can)d(solv)o(e)g(b)o(y)f(w)o(orking)g(in)h(\015oats.) 450 1917 y Fe(Exercise)j(1)450 2014 y Fp(Let)d(A)f(b)q(e)i(the)f(matrix)d (from)h(Exercise)j(2\(a\))e(of)g(Section)h(3.4.)j(Solv)o(e)c(the)h(system)f
(A)450 2064 y Fk(X)h Fp(=)g Fk(B)g Fp(where)512 2164 y Fk(B)20 b Fp(=)g([)643 2148 y Fi(1)p 635 2155 34 2 v 635 2178 a(10)673 2164 y Fp(])698 2081 y Ff(2)698 2155 y(4)763 2114 y Fp(21)763 2164 y(32)747 2213 y Fg(\000)p Fp(44)841 2081 y Ff(3)841 2155 y(5)869 2164 y Fp(,)h(then)g(con)o(v)o(ert)f(the)h(answ)o(er)g(to)f (\015oating)f(p)q(oin)o(t)g(form.)450 2264 y(Next,)11 b(con)o(v)o(ert)g(b)q (oth)f(A)g(and)g Fk(B)g Fp(to)g(\015oats)h(b)o(y)e(mapping)f(the)j(con)o(v)o (ert/\015oat)f(op)q(erator)450 2313 y(on)o(to)h(them.)17 b(Re-solv)o(e)11 b(the)h(system)f(and)h(compare)e(the)j(t)o(w)o(o)e(\015oating)f(p)q(oin)o(t)h (results.)450 2363 y(\(See)k(Exercise)g(4)f(in)f(Section)i(2.3.\))p eop bop 150 275 a Fm(3.4)13 b(Maple)h(On)g(Line)967 b(43)150 391 y Fe(Exercise)17 b(2)150 475 y Fp(In)h(man)o(y)e(applications)h(of)g(linear)h (algebra,)g(n)o(umerical)e(data)i(comes)f(from)f(mea-)150 525 y(suremen)o(ts)d(whic)o(h)f(are)g(susceptible)i(to)e(error.)19 b(Supp)q(ose)13 b(the)g(v)o(ector)f Fk(B)h Fp(in)e(Exercise)150 574 y(1)19 b(w)o(as)h(obtained)g(b)o(y)f(measuring)g(a)g(v)o(ector)i Fk(Ba)h Fp(whose)f(actual)e(v)n(alue)g(is)h Fk(Ba)g Fp(=)150 672 y([)183 655 y Fi(1)p 167 662 50 2 v 167 686 a(100)221 672 y Fp(])247 588 y Ff(2)247 663 y(4)311 622 y Fp(210)311 671 y(321)295 721 y Fg(\000)p Fp(440)410 588 y Ff(3)410 663 y(5)438 672 y Fp(.)e(Compute)13 b(the)h(solution)f(to)h(the)h(equation)e(A)h Fk(Xa)g Fp(=)g Fk(Ba.)212 773 y Fp(Measure)j(error)f(is)e(the)i(absolute)f(v) n(alue)f(of)h(the)g(di\013erence)i(b)q(et)o(w)o(een)g(the)e(com-)150 823 y(puted)i(v)n(alue)e(and)h(the)g(actual)f(v)n(alue.)24 b(It)16 b(can)g(b)q(e)g(computed)g(in)f(Maple)h(with)f(the)150 872 y(follo)o(wing)c(command.)212 949 y Fn(>)282 953 y Fo(e)21 b(:=)h(map\(abs,evalm\(X-)o(Xa\)\);)212 1082 y Fp(Whic)o(h)c(comp)q(onen)o(t) g(of)f(the)i(solution)f Fk(X)g Fp(computed)g(in)g(Exercise)i(1)e(has)g(the) 150 1132 y(largest)13 b(error?)19 b(\(It)13 b(migh)o(t)e(hepl)i(to)g(con)o(v) o(ert)h(y)o(our)f(answ)o(er)g(to)g(\015oting)g(p)q(oin)o(t)f(form.\))150 1182 y(In)i(terms)f(of)g(the)h(magnitude)e(of)i(the)g(comp)q(onen)o(ts)f(of)g (the)i(in)o(v)o(erse)f Fj(A)1274 1167 y Fi(\()p Fh(\000)p Fi(1\))1345 1182 y Fp(,)f(explain)150 1232 y(wh)o(y)19 b(this)f(is)h(the)g(largest)g(to)g (b)q(e)h(exp)q(ected.)34 b(\(Maple)19 b(computes)g(the)g(in)o(v)o(erse)h(of) 150 1281 y(a)d(matrix)e(with)i(the)g Fk(in)o(v)o(erse)e Fp(command.)24 b(\))k(Whic)o(h)17 b(comp)q(onen)o(t)f(of)h Fk(Ba)g Fp(w)o(ould)150 1331 y(y)o(ou)f(c)o(hange)h(in)f(order)h(to)g(pro)q(duce)h(the)f(greatest)h (c)o(hange)f(in)f Fk(Xa)p Fp(?)26 b(Wh)o(y?)g(Bac)o(k)150 1381 y(up)16 b(y)o(our)h(answ)o(er)g(with)f(a)g(n)o(umerical)f(example)g(or)h (with)g(a)g(sym)o(b)q(olic)f(calculation)150 1431 y(where)g(the)f(incremen)o (ts)g(in)g Fk(Ba)g Fp(are)g(parameters)g(successiv)o(ely)h(app)q(earing)f(in) f(eac)o(h)150 1481 y(comp)q(onen)o(t.)20 b(Ho)o(w)14 b(m)o(uc)o(h)g(error)i (could)e(y)o(ou)g(tolerate)i(in)e(the)h(measured)g(v)n(alues)g(of)150 1530 y(the)f(comp)q(onen)o(ts)f(of)g Fk(B)g Fp(if)g(the)h(absolute)g(v)n (alue)e(of)h(the)h(error)h(in)e(an)o(y)g(en)o(try)h(of)f Fk(X)g Fp(is)150 1580 y(to)h(b)q(e)g(at)g(most)f(.001?)150 1754 y Fe(Exercise)k(3)150 1838 y Fp(Let)d(A)g(and)g Fk(B)g Fp(b)q(e)g(as)g (de\014ned)h(b)q(elo)o(w.)212 1914 y Fn(>)282 1918 y Fo(A)21 b(:=)h(matrix\(3,3,[1,1/)o(2,1/3)o(,1/2,)o(1/3,1)o(/4,1/)o(3,1/)o(4,1/5)o (]\);)212 1994 y Fn(>)282 1998 y Fo(B)f(:=)h(vector\([83,46,32)o(]\);)212 2090 y Fp(Find)17 b(the)i(solution)d(to)i(A)f Fk(X)h Fp(=)g Fk(B)p Fp(.)e(As)i(in)g(Exercise)h(2,)e(supp)q(ose)i(the)f(v)o(ector)150 2139 y Fk(B)h Fp(w)o(as)g(obtained)f(b)o(y)h(measuring)f(a)h(v)o(ector)g Fk(Ba)g Fp(whose)h(actual)f(v)n(alue)f(is)h Fk(Ba)g Fp(=)150 2237 y([)183 2220 y Fi(1)p 167 2227 V 167 2251 a(100)221 2237 y Fp(])247 2153 y Ff(2)247 2228 y(4)295 2186 y Fp(8290)295 2236 y(4607)295 2286 y(3130)399 2153 y Ff(3)399 2228 y(5)426 2237 y Fp(.)j(Solv)o(e)15 b(the)h(equation)e(A)i Fk(Xa)f Fp(=)h
Fk(Ba)p Fp(.)22 b(What)14 b(is)h(the)h(p)q(ercen)o(tage)150 2334 y(error)f(in)f(the)h(least)f(accurate)i(en)o(try)f(of)e Fk(X)p Fp(?)h(Ho)o(w)g(m)o(uc)o(h)f(error)j(could)e(y)o(ou)g(tolerate)150 2384 y(in)f(the)h(measured)f(v)n(alues)h(of)e(the)i(comp)q(onen)o(ts)f(of)g Fk(B)h Fp(if)e(the)i(absolute)g(v)n(alue)f(of)f(the)150 2434 y(error)j(in)e(an)o(y)h(en)o(try)g(of)f Fk(X)h Fp(is)g(to)g(b)q(e)g(at)g (most)f(.001?)p eop bop 450 275 a Fp(44)967 b Fm(3.4)13 b(Maple)h(On)g(Line)450 391 y Fe(Exercise)j(4)450 468 y Fp(Exercises)d(2)d(and)h(3)f(demonstrate)h (that)f(the)i(pro)q(cess)g(of)e(solving)g(a)g(system)h(of)f(equa-)450 518 y(tions)16 b(can)g("magnify")d(errors)18 b(in)e(disastrous)g(w)o(a)o(ys.) 25 b(One)17 b(quan)o(titativ)o(e)e(measure)450 568 y(of)d(the)i(inaccuracy)f (of)g(a)f(calculation)g(is)h(the)g(ratio)g(of)f(the)h(p)q(ercen)o(tage)i (error)f(in)e(the)450 618 y(\014nal)17 b(answ)o(er)h(to)g(the)g(p)q(ercen)o (tage)h(error)g(in)e(the)h(input)g(data.)29 b(But)18 b(what)f(do)h(w)o(e)450 668 y(mean)13 b(b)o(y)h(the)g(p)q(ercen)o(tage)i(error)f(in)e(a)h(v)o(ector)h (\(suc)o(h)g(as)f Fk(X)g Fp(in)g(Exercises)i(1)d(and)h(2\))450 718 y(in)f(whic)o(h)h(ev)o(ery)h(comp)q(onen)o(t)e(migh)o(t)f(ha)o(v)o(e)h (errors)j(of)d(di\013eren)o(t)i(magnitudes?)512 768 y(F)m(or)f(v)o(ectors)g (in)g Fj(R)808 753 y Fi(3)826 768 y Fp(,)f(this)h(question)g(has)g(a)f (geometric)h(meaning.)i(Think)d(of)g Fj(X)1781 774 y Fi(1)450 817 y Fp(and)h Fj(X)565 823 y Fi(2)598 817 y Fp(as)g(represen)o(ting)i(p)q (oin)o(ts)e(in)g Fj(R)1090 802 y Fi(3)1108 817 y Fp(.)19 b(The)c(distance)g Fl(d)f Fp(b)q(et)o(w)o(een)h(these)h(p)q(oin)o(ts)450 867 y(is)g(one)g (measure)f(of)h(the)g(error.)25 b(If)15 b Fj(X)1059 873 y Fi(1)1094 867 y Fp(=)h([)p Fj(x)1178 873 y Fi(1)1196 867 y Fj(;)d(y)1241 873 y Fi(1)1260 867 y Fj(;)h(z)1305 873 y Fi(1)1324 867 y Fp(])1336 852 y Fn(t)1366 867 y Fp(and)h Fj(X)1482 873 y Fi(2)1517 867 y Fp(=)h([)p Fj(x)1601 873 y Fi(2)1619 867 y Fj(;)d(y)1664 873 y Fi(2)1683 867 y Fj(;)h(z)1728 873 y Fi(2)1747 867 y Fp(])1759 852 y Fn(t)1788 867 y Fp(,)450 917 y(then)738 1010 y Fj(d)d Fp(=)815 974 y Ff(p)p 856 974 645 2 v 856 1010 a Fp(\()p Fj(x)896 1016 y Fi(1)924 1010 y Fg(\000)e Fj(x)989 1016 y Fi(2)1008 1010 y Fp(\))1024 998 y Fi(2)1052 1010 y Fp(+)g(\()p Fj(y)1129 1016 y Fi(1)1157 1010 y Fg(\000)h Fj(y)1219 1016 y Fi(2)1238 1010 y Fp(\))1254 998 y Fi(2)1282 1010 y Fp(+)f(\()p Fj(z)1358 1016 y Fi(1)1387 1010 y Fg(\000)g Fj(z)1447 1016 y Fi(2)1466 1010 y Fp(\))1482 998 y Fi(2)1501 1010 y Fp(.)512 1102 y(In)i(Maple,)g(this)g (can)h(b)q(e)f(computed)g(as)g("norm\()e Fj(X)1314 1108 y Fi(1)1337 1102 y Fg(\000)s Fj(X)1406 1108 y Fi(2)1426 1102 y Fp(,)i(2\)".)16 b(The)c(additional)450 1152 y("2")e(represen)o(ts)k(the)e("2-norm")d(wherein) j(di\013erences)i(are)d(squared)h(and)f(the)h(square)450 1202 y(ro)q(ot)i(of)g(the)g(sum)f(tak)o(en.)19 b(If)14 b Fj(X)952 1208 y Fi(1)985 1202 y Fp(is)g(the)g(computed)g(answ)o(er)h(and)f Fj(X)1543 1208 y Fi(2)1576 1202 y Fp(is)g(the)g(actual)450 1252 y(answ)o(er,)g(w)o(e)g(de\014ne)h(the)g(p)q(ercen)o(tage)g(error)g(to)f (b)q(e)735 1344 y(P)g(=)g(100)f(norm\()g Fj(X)1058 1350 y Fi(1)1086 1344 y Fg(\000)d Fj(X)1162 1350 y Fi(2)1181 1344 y Fp(,)j(2\)/norm\()f Fj(X)1422 1350 y Fi(1)1441 1344 y Fp(,)i(2\).)512 1437 y(a\))g(Let)g Fk(B)p Fp(,)f Fk(Ba)p Fp(,)h Fk(X)g Fp(and)f Fk(Xa)h Fp(b)q(e)g(as)g(in)g (Exercise)h(2.)j(Use)c(the)h(giv)o(en)e(form)o(ula)e(to)450 1486 y(compute)j(\(i\))g(the)h(p)q(ercen)o(tage)i(error)e(in)f Fk(B)p Fp(,)g(\(ii\))g(the)h(p)q(ercen)o(tage)h(error)g(in)e Fk(X)p Fp(,)g(and)450 1536 y(\(iii\))g(the)j(ratio)e(of)g(the)h(p)q(ercen)o (tage)h(error)g(in)e Fk(X)h Fp(to)f(that)h(in)f Fk(B)p Fp(.)g(This)h(is)f (the)h(inac-)450 1586 y(curacy)f(of)e(the)i(calculation)e(of)g Fk(X)h Fp(from)e Fk(B)p Fp(.)i(Assuming)e(that)i(accuracy)h(is)f(desired,)450 1636 y(do)g(w)o(e)g(w)o(an)o(t)f(the)i(n)o(um)o(b)q(er)e Fl(d)h Fp(to)g(b)q(e)g(large)g(or)g(small?)h(Explain.)512 1686 y(b\))d(Compute)f
(the)h(inaccuracy)h(of)e(the)h(computation)e(of)h Fk(X)h Fp(from)e Fk(B)i Fp(in)f(Exercise)450 1736 y(3.)512 1786 y(c\))j(F)m(or)e(eac)o(h)i Fl(n)f Fp(x)g Fl(n)g Fp(in)o(v)o(ertible)f(matrix)f(A)i(there)i(is)d(a)h(n)o (um)o(b)q(er)f("cond\(A\)")h(\(the)450 1836 y("condition)i(n)o(um)o(b)q(er)g (of)g(A"\))h(suc)o(h)h(that)f(the)g(inaccuracy)h(in)e(solving)g(the)h(system) 450 1885 y(A)c Fk(X)h Fp(=)f Fk(B)h Fp(is)f(at)g(most)f(cond\(A\),)i (regardless)g(of)f Fk(B)g Fp(and)h(regardless)g(of)f(the)h(amoun)o(t)450 1935 y(of)j(error)i(in)f Fk(B)p Fp(.)f(This)h(means)f(that)h(if,)g(sa)o(y)m (,)g(cond\(A\))g(=)h(20)e(and)h(the)h(error)f(in)g Fk(B)450 1985 y Fp(is)i(.001\045,)f(then)h(the)g(computed)g(v)n(alue)f(of)g Fk(X)h Fp(will)e(ha)o(v)o(e)i(at)g(most)e(20)h(x)h(.001)f(=)450 2035 y(.02\045)11 b(error.)19 b(In)13 b(general,)g(1)f Fg(\024)20 b Fp(cond\(A\).)13 b(\(This)g(sa)o(ys)g(that)g(w)o(e)g(cannot)g(exp)q(ect)h (the)450 2085 y(answ)o(er)h(to)e(b)q(e)i(more)e(accurate)i(than)f(the)g (input)g(data.\))512 2135 y(Maple)19 b(has)h(a)f(built-in)f(command)f(for)i (the)h(condition)f(n)o(um)o(b)q(er.)33 b(Since)20 b(the)450 2185 y(condition)14 b(n)o(um)o(b)q(er)h(is)g(constructed)i(from)c(a)i("norm") e(of)h(the)i(matrix)d(A,)i(w)o(e)g(need)450 2234 y(to)h(sp)q(ecify)h(the)f(v) o(ersion)h(of)e(the)i(condition)f(n)o(um)o(b)q(er)f(w)o(e)h(w)o(an)o(t,)g (according)g(to)g(the)450 2284 y(t)o(yp)q(e)g(of)f(norm)f(w)o(e)i(w)o(an)o(t) f(used.)24 b(Hence,)16 b(w)o(e)g(will)e(use)j(the)f(syn)o(tax)f(cond\(A,2\).) 23 b(In)450 2334 y(addition,)11 b(the)i(2-norm)e(of)h(a)g(sym)o(b)q(olic)e (matrix)h(can)i(b)q(e)g(a)f(v)o(ery)h(large)f(and)g(complex)450 2384 y(expression.)28 b(F)m(or)16 b(this)h(reason,)h(w)o(e)f(will)e(only)h (compute)h(the)g(condition)f(n)o(um)o(b)q(er,)450 2434 y(based)f(on)e(the)i (2-norm,)c(of)j(matrices)f(of)g(\015oating)g(p)q(oin)o(t)h(n)o(um)o(b)q(ers.) p eop bop 150 275 a Fm(3.5)13 b(Maple)h(On)g(Line)967 b(45)212 391 y Fp(Compute)14 b(the)h(condition)e(n)o(um)o(b)q(ers)i(for)f(the)h(co)q (e\014cien)o(t)g(matrices)f(in)g(Exercise)150 441 y(1)j(and)g(3.)29 b(Use)18 b(this)g(to)f(explain)g(the)h(di\013erence)h(in)e(accuracies)i (obaine)e(in)h(these)150 491 y(exercises.)212 543 y(Matrices)11 b(with)e(large)g(condition)g(n)o(um)o(b)q(ers)g(are)h(called)f ("ill-conditioned.")e(If)i(the)150 593 y(co)q(e\016cien)o(t)k(matrix)e(of)g (a)i(system)f(is)g(ill-conditioned,)e(then)j(w)o(e)g(m)o(ust)e(b)q(e)i (extremely)150 643 y(suspicious)h(of)f(answ)o(ers)i(obtained)e(b)o(y)h (solving)e(the)i(system)g(since)g(an)o(y)f(sligh)o(t)g(error)150 692 y(in)g(the)g(input)g(data)g(can)g(mak)o(e)f(the)i(solution)e(v)o(ery)h (inaccurate.)19 b(Notice)13 b(that)h(these)150 742 y(inaccuracies)i(are)f (not)g(related)h(to)f(round-o\013)g(error.)22 b(Ill-conditioning)12 b(is)j(in)o(trinsic)150 792 y(in)e(the)i(matrix)d(and)i(not)f(in)h(the)g (metho)q(d)f(of)h(solution.)150 941 y Fq(3.5)66 b(On)22 b(Line)150 1035 y Fp(Restart)f(Maple,)h(clearing)f(its)g(memory)d(of)i(all)g(de\014ned)i (v)n(ariables.)38 b(Then,)22 b(re-)150 1085 y(initialize)12 b(b)o(y)i(loading)e(the)j Fl(linalg)e Fp(and)h Fl(student)g Fp(pac)o(k)n(ages.)212 1137 y(These)f(exercises)h(will)d(explore)h Fk(LUdecomp)p Fp(,)e(Maple's)i(built-in)e(command)f(for)150 1187 y(obtaining)17 b(the)h(LU)h(decomp)q(osition)d(of)i(a)g(matrix.)29 b(The)18 b(LUdecomp)g(command)150 1237 y(can)13 b(return)h(sev)o(en)g (di\013eren)o(t)f(items,)f(namely)m(,)e(L,)i(U,)h(a)f(factorization)g(of)g(U) h(in)o(to)f(U1)150 1287 y(and)h(R,)f(a)h(p)q(erm)o(utation)e(matrix)g(P)m(,)i (the)g(determinan)o(t)g(of)f(U1,)h(and)f(the)i(rank)f(of)f(A.)150 1336 y(W)m(e)h(will)g(not)g(need)i(all)e(of)g(these)i(outputs,)f(and)f(will)g (restrict)i(explorations)e(to)h(just)150 1386 y(L,)f(U,)h(and)g(P)m(.)212 1438 y(The)h(syn)o(tax)e(for)h(a)g(m)o(ulti-return)e(Maple)i(function)g(is)f (tedious.)19 b(Eac)o(h)14 b(v)n(ariable)150 1488 y(that)i(is)g(to)h(ha)o(v)o (e)f(a)g(return)h(assigned)g(to)f(it)g(m)o(ust)f(b)q(e)i(included)f(in)g(the) h(command)150 1538 y(surrounded)e(b)o(y)f(single)f(quotes.)212 1590 y(Since)j(the)f(actual)g(return)h(of)e(the)h Fk(LUdecomp)e
Fp(command)f(is)j(U,)g(w)o(e)g(will)e(not)150 1640 y(need)21 b(to)e(mak)o(e)f(a)h(sp)q(eci\014c)j(request)f(for)e(U)h(to)f(b)q(e)h (returned.)37 b(Th)o(us,)21 b(to)f(obtain)150 1689 y(L,)c(U,)f(and)h(an)o(y)g (p)q(erm)o(utation)f(P)h(needed)h(to)f(complete)g(the)h(decomp)q(osition,)d (the)150 1739 y(appropriate)g(syn)o(tax)g(w)o(ould)f(b)q(e)490 1841 y(u)h(:=)f(LUdecomp\(A,)g(L)h(=)g('l',)e(P)i(=)g('p'\);)212 1941 y(Of)j(course,)h(if)e(only)g(L)h(and)f(U)h(w)o(ere)h(desired,)g(then)f (the)h(command)c(could)i(b)q(e)150 1991 y(shortened)f(to)570 2093 y(u)e(:=)h(LUdecomp\(A,)f(L)h(=)g('l'\);)150 2253 y Fe(Exercise)j(1)150 2334 y Fp(En)o(ter)g(the)h(Matrix)e(A)g(from)f(Example)g(1)h(of)g(the)h (text,)g(then)g(obtain)f(the)h(LU)g(de-)150 2384 y(comp)q(osition)10 b(b)o(y)i(using)f(the)i Fk(LUdecomp)d Fp(command.)k(Assign)e(the)h(output)f (of)f(this)150 2434 y(command)i(to)i(the)h(v)n(ariable)e(u,)h(and)g(let)h(l)e (b)q(e)i(the)g(lo)o(w)o(er)f(triangular)g(factor.)22 b(This)p eop bop 450 275 a Fp(46)967 b Fm(3.5)13 b(Maple)h(On)g(Line)450 391 y Fp(matrix)f(will)f(not)j(need)g(a)f(p)q(erm)o(utation)f(P)m(,)h(so)g (it)g(can)h(b)q(e)g(omitted)e(from)f(the)j(com-)450 441 y(mand.)j(Finally)m (,)12 b(v)o(erify)i(that)h(A)g(=)f(l)g(u.)20 b(\(Y)m(ou)14 b(ma)o(y)f(need)i(to)g(use)g(either)g Fk(prin)o(t)d Fp(or)450 491 y Fk(ev)m(alm)i Fp(to)f(view)h(the)g(con)o(ten)o(ts)h(of)f(l.\))450 634 y Fe(Exercise)j(2)450 711 y Fp(Before)d(examining)c(the)k(consequences)i (piv)o(oting)11 b(has)i(on)g(the)h(LU)f(decomp)q(osition,)450 761 y(w)o(e)k(study)g(the)g(notion)f(of)h(a)f(matrix)f(factorization.)26 b(When)16 b(w)o(e)h(demand)f(that)h(A)450 811 y(b)q(e)h("factored")f(in)o(to) f(the)i(pro)q(duct)g(LU,)f(with)f(L)h(b)q(eing)g(a)g(lo)o(w)o(er-triangular,) f(and)450 860 y(U)j(b)q(eing)f(an)h(upp)q(er-triangular)f(matrix,)f(w)o(e)i (are)g(asking)f(for)g(the)i(solution)d(of)h(a)450 910 y(set)g(of)e (equations.)26 b(F)m(or)16 b(example,)g(if)g(A)h(is)f(a)h(giv)o(en)f(3)g(x)h (3)f(matrix,)f(then)i(\014nding)450 960 y(matrices)e(L)g(and)h(U)f(suc)o(h)i (that)e(A=LU)h(is)g(equiv)n(alen)o(t)e(with)i(solving)e(a)h(system)h(of)450 1010 y(nine)d(equations,)g(where)i(eac)o(h)f(equations)f(is)g(obtained)h(b)o (y)f(setting)h(one)f(en)o(try)h(of)f(A)450 1060 y(equal)h(to)f(the)i(corresp) q(onding)g(en)o(try)f(of)f(LU.)512 1110 y(In)g(this)h(exercise,)g(w)o(e)g(in) o(v)o(estigate)f(this)g(system)g(for)g(the)g(matrix)f(A)h(of)f(Exercise)450 1159 y(1.)18 b(Begin)c(b)o(y)f(forming)f(L)i(and)f(U,)h(3x3)f(matrices)h(of)f (indeterminates)h Fj(L)1607 1165 y Fn(ij)1650 1159 y Fp(and)g Fj(U)1759 1165 y Fn(ij)1788 1159 y Fp(.)512 1209 y(Maple's)d Fk(matrix)f Fp(command,)e(with)j(appropriate)g(if-statemen)o(ts)g(as)g(an)g (option,)450 1259 y(will)h(create)k(the)e(desired)h(matrices.)512 1324 y Fn(>)582 1328 y Fo(L)21 b(:=)h(matrix\(3,3,\(i,j\))o(->)d(if)i(i<j)g (then)g(0)h(else)f(L.i.j;)f(fi\);)h(U)512 1371 y(:)h(=matrix\(3,3,\(i,j\))o (->)d(if)i(i>j)g(then)g(0)h(else)f(U.i.j;fi\);)512 1447 y Fp(Multiply)12 b(L)h(and)g(U,)f(forming)f(a)h(template)g(of)h(indeterminates)f(whic)o(h)h(w) o(e)g(then)450 1497 y(demand)c(reduce)i(to)f(the)h(en)o(tries)g(of)e(A)h (from)e(Exercise)j(1,)f(ab)q(o)o(v)o(e.)17 b(This)9 b(giv)o(es)h(a)g(set)h (of)450 1547 y(nine)i(equations)f(in)g(t)o(w)o(elv)o(e)g(unkno)o(wns,)h (since)g(there)h(are)f(six)f(unkno)o(wns)h(in)f(eac)o(h)h(of)450 1597 y(L)e(and)g(U.)g(The)g(solution)g(of)f(this)h(set)h(of)f(equations)g (will)f(not)h(b)q(e)g(unique,)h(suggesting)450 1647 y(that)i(w)o(e)g(can)g (imp)q(ose)f(an)h(additional)e(three)j(conditions)e(on)h(the)g (factorization.)512 1712 y Fn(>)582 1716 y Fo(LU)21 b(:=)h(evalm\(L)e(&*)h (U\);)512 1793 y Fp(The)13 b(pro)q(duct)g(LU)f(and)g(the)h(matrix)d(A)i(can)g (b)q(e)h(equated)g(via)e(the)i Fk(equate)e Fp(com-)450 1843 y(mand)i(from)f(the)j Fl(student)f Fp(pac)o(k)n(age.)19 b(This)14 b(command)e(returns)j(a)f(set)h(of)f(equations)450 1892 y(formed)f(b)o(y)g (equating)h(corresp)q(onding)h(en)o(tries)g(of)e(eac)o(h)h(matrix.)512 1958 y Fn(>)582 1962 y Fo(q)21 b(:=)h(equate\(LU,A\);)512 2039 y Fp(T)o(ypically)m(,)c(the)h Fk(solv)o(e)f Fp(command)e(needs)k(a)f(set)g
(of)g(equations)g(and)f(a)h(set)h(of)450 2088 y(v)n(ariables.)d(If)c(no)g(v)n (ariables)f(are)i(suggested)g(to)f(Maple,)g(it)f(will)g(attempt)g(to)h (deduce)450 2138 y(what)j(the)h(unkno)o(wns)f(of)g(the)h(problem)d(actually)i (are.)25 b(That)16 b(is)g(con)o(v)o(enien)o(t)h(here)450 2188 y(since)h(it)e(w)o(ould)g(b)q(e)i(tedious)f(to)g(en)o(ter)h(the)f(names)f(of) h(all)e(the)j(v)n(ariables)e(in)h(these)450 2238 y(equations.)512 2303 y Fn(>)582 2307 y Fo(q1)k(:=)h(solve\(q\);)512 2384 y Fp(It)13 b(is)g(clear)h(that)f(w)o(e)g(did)g(not)g(get)h(a)e(unique)i (solution)e(for)h(the)g(en)o(tries)i(of)d(L)h(and)450 2434 y(U.)i(Careful)f(insp)q(ection)i(sho)o(ws)f(there)i(are)e(three)i (indeterminates)d(in)h(the)h(answ)o(er.)p eop bop 150 275 a Fm(3.5)13 b(Maple)h(On)g(Line)967 b(47)150 391 y Fp(This)10 b(b)q(ecomes)h(more)e(eviden)o(t)h(if)g(w)o(e)g(substitute)i (these)f(solutions)f(in)o(to)g(the)g(matrices)150 441 y(L)k(and)f(U.)212 522 y Fn(>)282 526 y Fo(L1)21 b(:=)h(subs\(q1,op\(L\)\);)c(U1)k(:=)f (subs\(q1,op\(U\)\);)150 714 y Fe(Exercise)c(3)150 801 y Fp(Change)e(the)h (de\014nition)f(of)g(the)h(matrix)e(L)h(used)i(in)d(Exercise)k(2.)k(Since)16 b(there)h(are)150 850 y(three)12 b(free)g(parameters)f(in)g(the)h(solution)e (for)h(the)g(factors)h(L)f(and)g(U,)f(c)o(ho)q(ose)i(to)f(ha)o(v)o(e)150 900 y(the)i(diagonal)d(elemen)o(ts)h(of)h(L)g(all)e(b)q(e)j(1's.)k(This)12 b(can)g(b)q(e)g(done)h(with)e(an)h(appropriate)150 950 y(if-statemen)o(t)g (in)h(the)h Fk(matrix)e Fp(command)e(that)k(de\014nes)g(L.)f(The)h(matrix)d (U)j(will)d(b)q(e)150 1000 y(the)18 b(same)f(as)h(used)g(in)f(Exercise)j(2.) 29 b(Rep)q(eat)18 b(the)g(formation)d(of)i(nine)h(equations)150 1050 y(in)e(nine)h(unkno)o(wns,)g(obtaining)e(unique)i(factors)g(L)f(and)h(U) f(for)h(the)g(matrix)e(A)h(in)150 1099 y(Exercise)e(1.)j(Displa)o(y)11 b(the)h(resulting)h(matrices)e(L)h(and)g(U,)g(and)g(sho)o(w)g(that)g(they)h (are)150 1149 y(exactly)h(the)g(factors)h(pro)q(duced)g(b)o(y)f(the)g Fk(LUdecomp)e Fp(command)f(in)j(Exercise)h(1.)150 1337 y Fe(Exercise)i(4)150 1423 y Fp(In)i(this)f(exercise)j(y)o(ou)d(will)f(explore)i(the)g(concept)h (of)e(a)g(p)q(erm)o(utation)f(matrix)g(P)150 1473 y(whose)j(ro)o(ws)g(are)h (a)e(p)q(erm)o(utation)g(of)g(the)i(ro)o(ws)f(of)f(the)h(iden)o(tit)o(y)g (matrix.)34 b(If)19 b(a)150 1523 y(matrix)12 b(A)j(is)f(m)o(ultiplied)e(b)o (y)i(P)m(,)f(the)i(ro)o(ws)g(of)f(P)m(A)g(will)f(b)q(e)i(p)q(erm)o(uted)f(in) g(the)h(same)150 1573 y(w)o(a)o(y)9 b(that)h(the)h(ro)o(ws)f(the)g(iden)o (tit)o(y)g(w)o(ere)g(when)h(forming)c(P)m(.)j(Let)g(P)g(b)q(e)h(a)e(p)q(erm)o (utation)150 1623 y(of)15 b(the)i(ro)o(ws)f(of)f(the)h(4x4)f(iden)o(tit)o(y)m (.)23 b(This)16 b(can)g(b)q(e)g(done)g(in)g(Maple)f(b)o(y)h(stac)o(king)f(a) 150 1672 y(sequence)k(of)d(ro)o(ws)h(from)f(the)h(iden)o(tit)o(y)m(.)26 b(Create)18 b(A,)f(a)f(random)g(4x4)g(matrix)f(and)150 1722 y(examine)e(A,)g(P)m(A,)g(and)h(P)m(.)212 1803 y Fn(>)282 1807 y Fo(Id)21 b(:=)h(diag\(1$4\);)212 1889 y Fn(>)282 1893 y Fo(P)f(:=)h (stack\(seq\(row\(Id)o(,k\),)c(k)k(=)g([2,4,3,1]\)\);)212 1974 y Fn(>)282 1978 y Fo(A)f(:=)h(randmatrix\(4,4\);)212 2059 y Fn(>)282 2063 y Fo(PA)f(:=)h(evalm\(P)e(&*)h(A\);)212 2144 y Fn(>)282 2148 y Fo(print\(A,PA,P\);)212 2246 y Fp(Finally)m(,)e(observ)o(e) j(that)e(for)g(a)g(p)q(erm)o(utation)f(matrix)f(P)m(,)i(the)g(in)o(v)o(erse)h (is)f(the)150 2296 y(transp)q(ose.)212 2378 y Fn(>)282 2382 y Fo(print\(inverse\(P\))o(,tran)o(spose)o(\(P\)\);)p eop bop 450 275 a Fp(48)967 b Fm(3.5)13 b(Maple)h(On)g(Line)450 391 y Fe(Exercise)j(5)450 468 y Fp(W)m(e)i(wish)g(to)h(study)g(the)g (e\013ect)h(of)e(piv)o(oting)f(on)h(the)h(LU)g(decomp)q(osition.)33 b(F)m(or)450 518 y(this,)19 b(w)o(e)g(need)h(a)e(matrix)f(A)i(that)g(forces)g (a)f(piv)o(ot.)32 b(If)18 b(the)i(\(1,1\)-elemen)o(t)d(of)h(A)450 568 y(w)o(ere)13 b(to)f(b)q(e)h(zero,)g(a)g(piv)o(ot)e(w)o(ould)h(ha)o(v)o(e) g(to)g(b)q(e)h(p)q(erformed)f(immedia)o(tely)m(,)d(since)k(the)450 617 y(ro)o(w-reduction)18 b(pro)q(cess)i(b)o(y)e(whic)o(h)g(w)o(e)g(obtain)f (L)h(and)g(U)g(is)g(basically)e(gaussian)450 667 y(elimination.)i(Create)e (A,)e(a)h(random)e(4x4)h(matrix,)f(and)h(then)i(re-assign)f(its)g(\(1,1\)-)
450 717 y(elemen)o(t)e(the)i(v)n(alue)e(zero.)512 767 y(Use)19 b(the)f Fk(LUdecomp)e Fp(command)e(to)k(obtain)f(the)h(LU)f(decomp)q (osition,)g(this)450 817 y(time)d(including)h(parameter)g(P)h(=)g('p')f(so)g (that)h(a)f(p)q(erm)o(utation)g(matrix)f(p)h(will)f(b)q(e)450 866 y(returned.)31 b(T)m(o)17 b(ha)o(v)o(e)h(Maple)f(displa)o(y)g(p,)h(l,)g (and)g(u)f(side-b)o(y-side,)i(use)f(the)h Fk(prin)o(t)450 916 y Fp(command.)512 981 y Fn(>)582 985 y Fo(u)i(:=)h(LUdecomp\(A,L='l')o(,P='p) o('\):)40 b(print\(p,l,u\);)512 1062 y Fp(Use)20 b(the)f Fk(prin)o(t)d Fp(command)f(to)k(displa)o(y)m(,)e(side-b)o(y-side,)j(the)f(pro)q(duct)h(lu,) e(the)450 1112 y(matrix)12 b(A,)i(and)f(the)i(pro)q(duct)g(plu.)512 1162 y(Y)m(ou)j(should)g(observ)o(e)i(that)f(the)g(pro)q(duct)g(lu)f(do)q(es) h(not)g(repro)q(duce)h(A.)e(That)450 1211 y(is)d(the)h(e\013ect)h(of)e(piv)o (oting.)20 b(Whenev)o(er)d(ro)o(ws)e(m)o(ust)g(b)q(e)g(in)o(terc)o(hanged)h (during)f(the)450 1261 y(factorization,)10 b(these)i(in)o(terc)o(hanges)g (are)f(recorded)h(in)e(the)h(matrix)e(P)m(.)h(The)h(resulting)450 1311 y(LU)j(factorization)f(is)g(then)i(a)e(factorization)g(of)g Fj(P)1253 1296 y Fi(\()p Fh(\000)p Fi(1\))1337 1311 y Fp(A,)g(not)h(A,)f (resulting)h(in)f(the)450 1361 y(equalit)o(y)g Fj(P)641 1346 y Fi(\()p Fh(\000)p Fi(1\))725 1361 y Fp(A)h(=)g(LU.)f(Th)o(us,)h(A)g(=)g (PLU,)g(not)g(just)g(LU.)p eop bop 150 705 a Fs(Chapter)31 b(4)150 916 y Fr(Orthogonalit)m(y)150 1159 y Fq(4.1)66 b(On)22 b(Line)150 1255 y Fp(Restart)14 b(Maple)f(to)g (clear)g(its)g(memory)e(of)i(all)f(de\014ned)i(v)n(ariables,)e(and)h (re-initialize)150 1305 y(b)o(y)h(loading)e(the)i Fl(linalg)p Fp(,)f Fl(plots)p Fp(,)h Fl(student)p Fp(,)f(and)h Fl(plotto)n(ols)f Fp(pac)o(k)n(ages.)212 1380 y(Giv)o(en)18 b(the)h(v)o(ectors)h Fj(Q)592 1386 y Fi(1)628 1380 y Fp(=)679 1321 y Ff(\024)722 1354 y Fg(\000)p Fp(1)738 1404 y(1)795 1321 y Ff(\025)817 1380 y Fp(and)f Fj(Q)936 1386 y Fi(2)972 1380 y Fp(=)1023 1321 y Ff(\024)1066 1354 y Fp(1)1066 1404 y(2)1107 1321 y Ff(\025)1129 1380 y Fp(,)g(w)o(e)g(will)e(generate)i(a)150 1452 y(co)q(ordinate)d(grid)f (corresp)q(onding)h(to)f(a)h(space)g(in)f(whic)o(h)g Fj(Q)1115 1458 y Fi(1)1149 1452 y Fp(and)h Fj(Q)1265 1458 y Fi(2)1298 1452 y Fp(are)g(the)g(ba-)150 1502 y(sis)k(v)o(ectors.)36 b(W)m(e)20 b(will)e(plot)h(the)h(sk)o(ew)o(ed)h(grid)e(lines)g(in)h(red,)h(atop)e(a)h (standard)150 1552 y(co)q(ordinate)14 b(system)g(in)f(blac)o(k.)212 1625 y Fn(>)282 1629 y Fo(Q1)21 b(:=)h(vector\([-1,1]\);)c(Q2)k(:=)f (vector\([1,2]\);)212 1716 y Fp(The)g(v)o(ector)f(equation)g(of)f(a)h(line)g (through)g(the)g(tip)g(of)f(the)i(v)o(ector)g Fj(Q)1395 1722 y Fi(1)1433 1716 y Fp(and)150 1766 y(parallel)14 b(to)h(the)h(v)o(ector)g Fj(Q)583 1772 y Fi(2)616 1766 y Fp(is)f(giv)o(en)g(b)o(y)g Fk(R)p Fp(\(t\))g(=)g Fj(Q)1007 1772 y Fi(1)1041 1766 y Fp(+)g(t)g Fj(Q)1152 1772 y Fi(2)1171 1766 y Fp(.)21 b(T)m(o)15 b(form)e(parallel)150 1816 y(lines)e(through)g(the)h(tip)e(of)h(2)f Fj(Q)635 1822 y Fi(1)654 1816 y Fp(,)h(3)f Fj(Q)741 1822 y Fi(1)760 1816 y Fp(,)h(etc.,)h(w)o(e)f(need)h(to)f(form)e(the)i(v)o(ectors)h(k)f Fj(Q)1438 1822 y Fi(1)1468 1816 y Fp(+)150 1866 y(t)i Fj(Q)212 1872 y Fi(2)231 1866 y Fp(,)g(with)f(k)h(an)g(in)o(teger)h(in)f(some)f(in)o (terv)n(al)g([-a,a].)k(Alternativ)o(ely)m(,)c(the)i(equations)150 1915 y(of)h(lines)h(parallel)e Fj(Q)478 1921 y Fi(1)512 1915 y Fp(through)i(the)g(tip)f(of)g Fj(Q)892 1921 y Fi(2)911 1915 y Fp(,)g(2)g Fj(Q)1007 1921 y Fi(2)1026 1915 y Fp(,)g(etc,)i(w)o(e)f(for)f (the)h(v)o(ectors)h(k)150 1965 y Fj(Q)183 1971 y Fi(2)215 1965 y Fp(+)c(t)h Fj(Q)323 1971 y Fi(1)341 1965 y Fp(.)k(In)13 b(ev)o(ery)h(case)g (w)o(e)g(let)f(t,)g(the)h(parameter)f(of)g(the)h(line,)e(range)i(o)o(v)o(er)f (an)150 2015 y(in)o(terv)n(al)g([-b,b].)212 2067 y(Maple's)g(seq)h(command)c (can)j(b)q(e)g(used)h(to)f(generate)h(an)f(appropriate)g(sequence)150 2117 y(of)d(v)o(ector)i(represen)o(tations)h(of)d(the)i(grid)e(lines.)17 b(Plotting)10 b(them)g(in)h(red)h(will)d(pro)q(duct)150 2167
y(the)14 b(desired)i(grid.)212 2220 y(A)e(template)f(for)h(the)g(v)o(ector)h (form)d(of)h(eac)o(h)i(set)f(of)g(grid)f(lines)h(is)g(obtained)f(as)212 2293 y Fn(>)282 2297 y Fo(P1)21 b(:=)h(evalm\(k*Q1+t*Q2)o(\);)d(P2)i(:=)h (evalm\(k*Q2+t*Q1)o(\);)212 2384 y Fp(Sequences)c(of)d(equations)h(of)f(the)h (sk)o(ew)g(grid)f(lines)g(are)h(formed)f(with)g(the)h Fk(seq)150 2434 y Fp(command.)804 2539 y(49)p eop bop 450 275 a Fp(50)967 b Fm(4.1)13 b(Maple)h(On)g(Line)512 387 y Fn(>)582 391 y Fo(s1)21 b(:=)h(seq\([P1[1],P1[2)o(],t=-)o(5..5])o (,k=-5)o(..5\):)40 b(s2)21 b(:=)h(seq\([P2[1],P2[2])o(,t=-5)o(..5])o(,k=-5)o (..5\):)512 470 y Fp(The)14 b(plot)e(of)h(the)g(grid)g(lines)g(is)f(assigned) i(to)f(the)g(v)n(ariable)f(f1)g(so)h(that)h(it)e(can)h(b)q(e)450 519 y(used)k(again)d(in)h(one)h(more)f(activit)o(y)m(.)22 b(W)m(e)15 b(include)h(the)h(scaling)e(parameter)g(in)g(the)450 569 y Fk(plot)f Fp(command.)21 b(A)16 b(1-1)g(scaling)f(can)h(also)f(b)q(e)i(imp)q (osed)e(in)o(teractiv)o(ely)g(from)f(the)450 619 y(to)q(olbar.)j(A)d(view)g (windo)o(w)f(is)h(also)f(set)i(in)e(the)i Fk(plot)d Fp(command.)512 686 y Fn(>)582 690 y Fo(f1)21 b(:=)h(plot\()p Fg(f)p Fo(s1,s2)p Fg(g)p Fo(,)d(color=red,)g(scaling=constrained)o(,)g(view=[-6..6,-6..6)o (]\):)512 732 y(f1;)512 810 y Fp(F)m(or)e(a)h(\014nishing)f(touc)o(h,)h(w)o (e)g(use)g(the)h Fk(arro)o(w)e Fp(command)d(from)i(the)i Fl(plotto)n(ols)450 860 y Fp(pac)o(k)n(age)9 b(to)h(dra)o(w)g Fj(Q)779 866 y Fi(1)807 860 y Fp(and)f Fj(Q)916 866 y Fi(2)935 860 y Fp(,)h(the)g(basis)g(v)o(ectors) h(of)e(the)h(sk)o(ew)h(co)q(ordinate)f(system.)450 910 y(W)m(e)15 b(plot)g(the)i(\014rst)f(in)f(green)i(and)f(the)g(second)h(in)e(blue,)h(sup)q (erimp)q(osing)e(b)q(oth)i(on)450 960 y(the)e(sk)o(ew)o(ed)h(red)g(grid.)512 1026 y Fn(>)582 1030 y Fo(a1)21 b(:=)h(arrow\([0,0],[1,)o(2],)d (.2,.4,.2,color=g)o(reen)o(\):)41 b(a2)21 b(:=)g(arrow\([0,0],[-1,1],)o (.2,.4)o(,.2,c)o(olor=)o(blue\))o(:)512 1073 y(display\([f1,a1,a2]\))o(;)450 1219 y Fe(Exercise)c(1)450 1297 y Fp(Mo)q(dify)c(the)i(commands)d(in)i(the)h (In)o(tro)q(duction)f(to)g(pro)q(duce)i(red)f(grid)f(lines)g(corre-)450 1369 y(sp)q(onding)g(to)g(the)g(basis)h(v)o(ectors)g Fj(Q)1026 1375 y Fi(1)1058 1369 y Fp(=)1105 1310 y Ff(\024)1147 1344 y Fp(1)1147 1393 y(2)1189 1310 y Ff(\025)1225 1369 y Fp(and)f Fj(Q)1339 1375 y Fi(2)1371 1369 y Fp(=)1418 1310 y Ff(\024)1476 1344 y Fp(1)1460 1393 y Fg(\000)p Fp(2)1534 1310 y Ff(\025)1556 1369 y Fp(.)k(In)d(addition,)450 1441 y(pro)q(duce)20 b(a)f(graph)f(sho)o (wing)h(the)g(sk)o(ew)o(ed)h(grid)e(lines)h(and)g(the)g(basis)g(v)o(ectors)h (in)450 1491 y(green)14 b(and)f(blue.)k(Assign)d(this)f(graph)f(to)h(a)g(v)n (ariable)f(so)h(that)g(it)f(can)h(b)q(e)h(re-used)g(in)450 1541 y(Exercise)i(4.)450 1688 y Fe(Exercise)h(2)450 1765 y Fp(The)g(curv)o(e)h(de\014ned)g(implicitly)c(b)o(y)i(the)i(equation)e Fj(x)1318 1750 y Fi(2)1348 1765 y Fg(\000)1396 1747 y Fn(y)1414 1734 y Fd(2)p 1396 1755 35 2 v 1405 1779 a Fi(4)1452 1765 y Fp(=)g(1)h(is)f(a)h(h)o(yp)q(erb)q(ola)450 1815 y(that)d(will)e(b)q(e)i (plotted)g(in)g(Exercise)h(3.)j(Here,)c(sho)o(w)g(that)g(in)f(the)h(basis)g (of)f(Exercise)450 1865 y(1)h(\(with)f(co)q(ordinates)i Fl(u)f Fp(and)g Fl(v)p Fp(\))g(the)g(equation)g(of)f(this)h(h)o(yp)q(erb)q(ola)g(is) g(4)7 b Fj(u)g(v)12 b Fp(=)g(1.)512 1915 y(Begin)i(b)o(y)g(deducing)g(the)h (equations)e(of)h(the)g(transformation.)i(The)e(p)q(oin)o(t)g(ma-)450 1987 y(trix)f(M)g(is)f(constructed)j(with)e(columns)f Fj(Q)1131 1993 y Fi(1)1162 1987 y Fp(and)h Fj(Q)1275 1993 y Fi(2)1293 1987 y Fp(.)18 b(Letting)13 b Fk(X)g Fp(=)1562 1929 y Ff(\024)1604 1962 y Fj(x)1605 2012 y(y)1649 1929 y Ff(\025)1684 1987 y Fp(and)f Fk(U)450 2091 y Fp(=)496 2033 y Ff(\024)539 2066 y Fj(u)540 2116 y(v)583 2033 y Ff(\025)605 2091 y Fp(,)i(the)g(transformation)e (equations)i(are)g Fk(X)g Fp(=)g(M)g Fk(U)p Fp(.)512 2164 y(The)d(v)o(ectors) g Fk(X)f Fp(and)g(M)g Fk(U)f Fp(can)i(b)q(e)f(equated)h(with)f(the)g Fk(equate)f Fp(command)f(from)450 2214 y(the)16 b(studen)o(t)h(pac)o(k)n (age.)23 b(The)16 b(equation)f Fk(X)g Fp(=)h(M)g Fk(U)f Fp(de\014nes)i(the)f
(c)o(hange)g(of)f(basis)450 2264 y(from)d Fl(xy)p Fp(-co)q(ordinates)j(to)e Fl(uv)p Fp(-co)q(ordinates.)19 b(Once)c(these)h(equations)d(are)i(obtained,) 450 2313 y(Maple's)g Fk(subs)e Fp(command)g(can)i(b)q(e)h(used)g(to)e(imp)q (ose)g(this)h(c)o(hange)h(of)e(co)q(ordinates)450 2363 y(on)g(the)g(equation) g(of)f(the)h(h)o(yp)q(erb)q(ola.)k(An)c(appropriate)g(syn)o(taxt)g(migh)o(t)e (b)q(e)512 2430 y Fn(>)582 2434 y Fo(q1)21 b(:=)h(equate\(X,)d(MU\);)p eop bop 150 275 a Fm(4.2)13 b(Maple)h(On)g(Line)967 b(51)212 387 y Fn(>)282 391 y Fo(q2)21 b(:=)h(x^2)f(-)g(y^2/4)g(=)h(1;)212 458 y Fn(>)282 462 y Fo(q3)f(:=)h(subs\(q1,q2\);)212 530 y Fn(>)282 534 y Fo(q4)f(:=)h(simplify\(q3\);)150 682 y Fe(Exercise)17 b(3)150 760 y Fp(Use)j(Maple's)e Fk(implicit)o(pl)o(ot)e Fp(command)g(from)h (the)i Fl(plots)g Fp(pac)o(k)n(age)f(to)h(obtain)f(a)150 810 y(graph,)c(in)f(the)i(xy-co)q(ordinates)g(system,)e(of)h(the)h(original)d(h)o (yp)q(erb)q(ola.)19 b(Assign)14 b(the)150 860 y(graph)g(to)h(a)f(v)n(ariable) f(so)i(that)g(in)f(Exercise)i(4)e(it)g(can)h(b)q(e)g(re-used.)22 b(Be)15 b(sure)h(to)e(use)150 909 y(1-1)f(scaling)g(so)h(that)g(no)g (distortion)f(is)h(in)o(tro)q(duced)h(b)o(y)f(the)g(computer)g(screen.)212 977 y Fn(>)282 981 y Fo(g3)21 b(:=)h(implicitplot\(q2)o(,x=-1)o(0..10)o (,y=-1)o(0..10)o(,col)o(or=bl)o(ack,)c(scaling=constrained)o(\):)212 1023 y(g3;)150 1171 y Fe(Exercise)f(4)150 1249 y Fp(Use)e(Maple's)e Fk(displa)o(y)e Fp(command)g(to)i(sup)q(erimp)q(ose)h(on)f(the)h(sk)o(ew)o (ed)h(grid)e(lines)h(of)150 1299 y(Exercise)i(1,)e(the)h(graph)f(of)f(the)i (h)o(yp)q(erb)q(ola)g(from)d(Exercise)k(3.)j(Then)c(use)g(Maple's)150 1349 y(digitizer)d(\(clic)o(k)h(on)g(the)g(graph,)f(clic)o(k)h(on)g(a)f(p)q (oin)o(t)g(in)h(the)g(graph,)g(read)g(the)g(co)q(ordi-)150 1399 y(nates)g(in)f(the)h(windo)o(w)e(at)i(the)g(top-left)f(of)f(the)i (graphics)g(to)q(olbar\))f(to)g(appro)o(ximate)150 1449 y(the)20 b(co)q(ordinates)f(of)g(some)f(p)q(oin)o(t)h(on)g(the)g(h)o(yp)q(erb)q(ola.) 34 b(By)19 b(coun)o(ting)g(and)g(esti-)150 1498 y(mating,)c(infer)h(the)i (corresp)q(onding)f(uv-co)q(ordinates.)27 b(Sho)o(w)16 b(that)h(to)f(within)g (the)150 1548 y(accuracy)h(of)e(the)i(digitizer,)f(y)o(our)g(co)q(ordinates)g (satisfy)g(the)h(equation)e(of)h(the)g(h)o(y-)150 1598 y(p)q(erb)q(ola)e(in)f (the)i Fl(xy)p Fp(-system)f(and)g(in)f(the)i Fl(uv)p Fp(-system.)212 1648 y(F)m(or)j(example,)f(the)h(p)q(oin)o(t)g(\(2,4\))f(app)q(ears)h(to)g(b) q(e)g(almost)e(on)i(the)g(h)o(yp)q(erb)q(ola.)150 1698 y(The)e(digitizer)g (giv)o(es)f(\(2,)h(3.5\).)22 b(The)16 b(same)f(p)q(oin)o(t)g(app)q(ears)i(to) e(b)q(e)i(\(2,)e(0.1\))g(in)g(the)150 1748 y(sk)o(ew)f(grid.)k(Hence,)212 1815 y Fn(>)282 1819 y Fo(subs\(x=2,y=3.5,q)o(2\);)212 1886 y Fn(>)282 1890 y Fo(subs\(u=2,v=.1,q4)o(\);)150 2089 y Fq(4.2)89 b(On)22 b(Line)150 2181 y Fp(Restart)f(Maple,)h(clearing)f(its)g(memory)d(of) i(all)g(de\014ned)i(v)n(ariables.)38 b(Then,)22 b(re-)150 2231 y(initialize)12 b(b)o(y)i(loading)e(the)j Fl(linalg)e Fp(and)h Fl(student)g Fp(pac)o(k)n(ages.)212 2281 y(Y)m(ou)g(are)g(w)o(orking)f(for)h (an)g(engineering)g(\014rm)f(and)h(y)o(our)f(b)q(oss)i(insists)f(that)g(y)o (ou)150 2331 y(\014nd)g(one)g(single)g(solution)f(to)g(the)i(follo)o(wing)c (system:)580 2426 y(2)c Fj(x)i Fp(+)g(3)e Fj(y)k Fp(+)f(4)d Fj(z)k Fp(+)e(3)e Fj(w)12 b Fp(=)g(12)p Fj(:)p Fp(9)p eop bop 450 275 a Fp(52)967 b Fm(4.2)13 b(Maple)h(On)g(Line)874 391 y Fp(4)7 b Fj(x)i Fp(+)h(7)d Fj(y)j Fg(\000)g Fp(6)d Fj(z)k Fg(\000)e Fp(8)e Fj(w)12 b Fp(=)g Fg(\000)p Fp(7)p Fj(:)p Fp(1)880 484 y(6)7 b Fj(x)i Fp(+)g(10)e Fj(y)k Fg(\000)e Fp(2)e Fj(z)k Fg(\000)f Fp(5)d Fj(w)12 b Fp(=)g(5)p Fj(:)p Fp(9)512 577 y(Y)m(ou)i(ob)r (ject,)g(noting)f(that)512 628 y(\(a\))e(The)g(system)g(is)g(clearly)g (inconsisten)o(t:)17 b(the)11 b(sum)f(of)g(the)i(\014rst)g(t)o(w)o(o)e (equations)450 677 y(con)o(tradicts)15 b(the)f(third.)512 728 y(\(b\))c(Y)m(ou)f(need)h(at)f(least)g(four)g(equations)h(to)f(determine)g (four)g(unkno)o(wns)g(uniquely)m(.)450 777 y(Ev)o(en)14 b(if)f(the)i(system)f (w)o(ere)g(solv)n(able,)f(y)o(ou)g(couldn't)h(pro)q(duce)h(just)f(one)g
(solution.)512 828 y(The)d(b)q(oss)g(w)o(on't)f(tak)o(e)g("no")g(for)g(an)g (answ)o(er.)18 b(Concerning)10 b(\(a\),)h(the)g(b)q(oss)g(p)q(oin)o(ts)450 877 y(out)17 b(that)g(the)h(system)f(w)o(as)g(obtained)f(from)g(measured)h (data)g(and)f(an)o(y)h(inconsis-)450 927 y(tencies)i(can)e(only)g(b)q(e)g (due)h(to)f(exp)q(erimen)o(tal)g(error.)29 b(Indeed,)18 b(if)f(an)o(y)f(one)i (of)e(the)450 977 y(constan)o(ts)h(on)e(the)i(righ)o(t)e(sides)h(of)g(the)g (equations)g(w)o(ere)h(mo)q(di\014ed)d(b)o(y)h(.1)h(units)f(in)450 1027 y(the)f(appropriate)g(direction,)g(the)h(system)e(w)o(ould)g(b)q(e)i (consisten)o(t.)512 1077 y(Concerning)g(\(b\),)g(the)g(b)q(oss)g(sa)o(ys)g ("Do)f(the)h(b)q(est)h(y)o(ou)e(can.)20 b(W)m(e)15 b(will)e(pass)i(this)450 1127 y(data)c(on)g(to)h(our)f(customers)h(and)g(they)g(w)o(ouldn't)e(kno)o(w) h(what)g(to)h(do)f(with)g(m)o(ultiple)450 1177 y(answ)o(ers.")512 1227 y(After)16 b(some)f(though)o(t,)g(y)o(ou)g(realize)g(that)h(pro)r (jections)g(can)f(help)h(with)f(the)h(in-)450 1277 y(consistency)f(problem)e (The)h(giv)o(en)g(system)f(can)h(b)q(e)h(written)f(in)g(v)o(ector)g(format)e (as)576 1418 y Fj(x)613 1334 y Ff(2)613 1409 y(4)662 1368 y Fp(2)662 1417 y(4)662 1467 y(6)703 1334 y Ff(3)703 1409 y(5)740 1418 y Fp(+)e Fj(y)817 1334 y Ff(2)817 1409 y(4)887 1368 y Fp(3)887 1417 y(7)866 1467 y(10)928 1334 y Ff(3)928 1409 y(5)965 1418 y Fp(+)f Fj(z)1041 1334 y Ff(2)1041 1409 y(4)1106 1368 y Fp(4)1090 1417 y Fg(\000)p Fp(6)1090 1467 y Fg(\000)p Fp(2)1164 1334 y Ff(3)1164 1409 y(5)1201 1418 y Fp(+)g Fj(w)1287 1334 y Ff(2)1287 1409 y(4)1351 1368 y Fp(3)1335 1417 y Fg(\000)p Fp(8)1335 1467 y Fg(\000)p Fp(5)1409 1334 y Ff(3)1409 1409 y(5)1448 1418 y Fp(=)1492 1334 y Ff(2)1492 1409 y(4)1546 1368 y Fp(12)p Fj(:)p Fp(9)1540 1417 y Fg(\000)p Fp(7)p Fj(:)p Fp(1)1557 1467 y(5)p Fj(:)p Fp(9)1647 1334 y Ff(3)1647 1409 y(5)512 1558 y Fp(Y)m(ou)j(realize)h(that)f(this)h(system)f(w)o(ould)g(b)q(e)h(solv)n (able)e(if)h(the)h(v)o(ector)g(on)f(the)h(righ)o(t)450 1608 y(side)k(of)e(the)i(ab)q(o)o(v)o(e)f(equation)g(w)o(ere)h(in)e(the)i(space)g (spanned)g(b)o(y)f(the)h(four)f(v)o(ectors)450 1658 y(on)f(the)h(left.)22 b(Using)15 b(Maple's)g(rank)h(command,)c(y)o(ou)j(quic)o(kly)f(compute)h(as)g (2,)g(the)450 1708 y(rank)f(of)f(the)i(system)e(matrix)f(A,)885 1849 y(A)i(=)976 1765 y Ff(2)976 1840 y(4)1024 1798 y Fp(2)62 b(3)c(4)73 b(3)1024 1848 y(4)62 b(7)41 b Fg(\000)p Fp(6)h Fg(\000)p Fp(8)1024 1898 y(6)f(10)g Fg(\000)p Fp(2)h Fg(\000)p Fp(5)1338 1765 y Ff(3)1338 1840 y(5)512 1989 y Fp(sho)o(wing)12 b(that)h(these)h(four)e (v)o(ectors)i(in)e(fact)g(span)h(a)f(plane.)18 b(Call)11 b(this)h(plane)h(W.) 512 2087 y(Y)m(our)18 b(idea)f(is)h(to)f(let)h Fk(Bw)g Fp(b)q(e)h(the)f(pro)r (jection)g(of)f Fk(B)h Fp(=)1449 2003 y Ff(2)1449 2078 y(4)1503 2037 y Fp(12)p Fj(:)p Fp(9)1497 2086 y Fg(\000)p Fp(7)p Fj(:)p Fp(1)1513 2136 y(5)p Fj(:)p Fp(9)1603 2003 y Ff(3)1603 2078 y(5)1649 2087 y Fp(on)o(to)f(W.)450 2184 y(Since)h(the)h(system)e(is)h(so)g (nearly)g(consisten)o(t,)h Fk(Bw)f Fp(should)g(b)q(e)g(v)o(ery)h(close)f(to)g Fk(B)p Fp(.)450 2234 y(F)m(urthermore,)c(the)h(system)f(A)h Fk(X)f Fp(=)h Fk(Bw)f Fp(should)h(certainly)f(b)q(e)h(solv)n(able)f(and)g (one)450 2284 y(of)f(the)i(solutions)e(should)h(b)q(e)g(what)g(the)h(b)q(oss) f(is)g(lo)q(oking)e(for.)512 2334 y(Y)m(our)20 b(p)q(oin)o(t)f(\(b\))h(will)f (require)i(some)e(further)i(though)o(t.)35 b(Ho)o(w)o(ev)o(er,)22 b(y)o(ou)d(do)450 2384 y(ev)o(en)o(tually)d(come)g(up)g(with)h(an)f(idea)g (whic)o(h)h(will)e(b)q(e)i(describ)q(ed)h(in)f(the)g(exercises)450 2434 y(whic)o(h)d(follo)o(w.)p eop bop 150 275 a Fm(4.2)13 b(Maple)h(On)g(Line)967 b(53)212 391 y Fp(Before)18 b(going)d(on)h(to)g(the)h(exercises,)i(it)d(will)e(b)q(e)j (useful)g(to)f(en)o(ter)i(the)f(data)f(of)150 441 y(this)g(system)h(of)e (equations.)26 b(En)o(ter)17 b(the)g(matrix)e(A)h(and)h(the)g(v)o(ector)g Fk(B)p Fp(.)f(Con)o(v)o(ert)150 491 y Fk(B)f Fp(to)h(exact)g(rational)e(form) f(and)j(call)e(that)i(v)o(ector)g Fk(b)p Fp(.)22 b(Compute)14 b(the)i(rank)g(of)e(A,)150 541 y(v)o(erifying)f(that)h(it)f(is)h(indeed)g(2.)
212 613 y Fn(>)282 617 y Fo(A)21 b(:=)h(matrix\(3,4,[2,3,)o(4,3,4)o(,7,-6)o (,-8,6)o(,10,-)o(2,-5)o(]\);)212 688 y Fn(>)282 692 y Fo(B)f(:=)h (vector\([12.9,-7.)o(1,5.9)o(]\);)212 764 y Fn(>)282 768 y Fo(b)f(:=)h(map\(convert,B,ra)o(tiona)o(l\);)212 840 y Fn(>)282 844 y Fo(rank\(A\);)150 1006 y Fe(Exercise)17 b(1)150 1086 y Fp(Find)9 b(an)g(orthogonal)f(basis)i(for)f(the)g(column)f(space)j(of)d(A,) h(use)h(the)g(F)m(ourier)f(Theorem)150 1136 y(to)15 b(obtain)g Fk(Bw)p Fp(,)h(the)h(pro)r(jection)f(of)f Fk(B)g Fp(on)o(to)g(W,)g(the)h (column)e(space)j(of)e(A.)g(Then)150 1186 y(solv)o(e)f(A)h Fk(X)f Fp(=)h Fk(Bw)p Fp(,)g(expressing)h(the)f(solution)e(in)h(parametric)g (form.)k(W)m(rite)c Fk(X)h Fp(as)150 1236 y(a)g(sum)g(of)g(a)g("translation") f(v)o(ector,)j(and)e(v)o(ectors)i(in)e(the)h(n)o(ull)f(space)h(of)f(A.)g(Sho) o(w)150 1286 y(that)f(the)h(translation)f(v)o(ector)h(Maple)f(\014nds)g(is)h (not)f(orthogonal)f(to)h(the)h(n)o(ull)e(space)150 1335 y(of)g(A.)212 1388 y(First,)i(obtain)f(an)g(orthonormal)e(basis)j(for)f(the)h(span)g(of)f (the)h(column)e(space)j(of)150 1437 y(A.)i(Since)g(the)h(columns)d(of)i(A)g (are)g(not)g(linearly)f(indep)q(enden)o(t)i(\(A)g(has)f(rank)g(2\),)150 1487 y(determine)d(\(via)f Fk(rref)p Fp(\))f(whic)o(h)i(t)o(w)o(o)f(columns)f (of)i(A)f(to)h(tak)o(e)g(as)f(indep)q(enden)o(t,)i(and)150 1537 y(pass)g(those)h(t)o(w)o(o)e(v)o(ectors)i(to)f(Maple's)f Fk(GramSc)o(hmidt)d Fp(command)h(for)j(orthogo-)150 1587 y(nalization.)21 b(This)15 b(command)d(returns)17 b(a)e(list)g(of)f Fl(lists)h Fp(\(not)g(v)o(ectors)p 1126 1594 128 2 v 1 w(\),)g(so)h(map)d(the)150 1637 y(con)o(v)o(ert-to-v)o(ector)f(op)q(erator)g(on)o(to)f(this)h(output.)17 b(Y)m(ou)11 b(will)f(no)o(w)h(ha)o(v)o(e)g(a)h(list)f(of)f(t)o(w)o(o)150 1686 y(orthogonal)j(v)o(ectors.)19 b(An)14 b(appropriate)g(syn)o(tax)g(migh)o (t)d(b)q(e:)212 1758 y Fn(>)282 1762 y Fo(q)21 b(:=)h(map\(convert,Gram)o (Schmi)o(dt\([c)o(ol\(A,)o(1\),co)o(l\(A,)o(2\)]\),)c(vector\);)212 1848 y Fp(Next,)e(apply)e(the)h(F)m(ourier)g(Theorem)f(to)h(get)g Fk(Bw)p Fp(,)g(the)h(pro)r(jection)f(of)g Fk(B)f Fp(on)o(to)150 1898 y(W,)i(the)h(column)e(space)j(of)e(A.)h(W)m(ork)e(with)i Fk(b)p Fp(,)f(the)i(exact)f(v)o(ersion)g(of)f(the)h(v)o(ector)150 1947 y Fk(B)p Fp(.)d(The)g(form)o(ula)e(in)i(the)h(F)m(ourier)f(Theorem)g(is) g(implemen)o(ted)e(in)i(Maple)g(m)o(uc)o(h)f(as)150 1997 y(it)i(is)h(written) g(mathematicall)o(y)m(.)k(Reference)d(the)g(basis)e(v)o(ectors)i(as)f(q[1])e (and)i(q[2],)150 2047 y(compute)g(dot)f(pro)q(ducts)j(with)d(Maple's)h Fk(dotpro)q(d)e Fp(command,)f(and)j(norms)f(with)150 2097 y(Maple's)f Fk(norm)e Fp(command,)f(b)q(eing)j(sure)h(to)e(compute)h(the)g(2-norm.)212 2149 y(Next,)19 b(solv)o(e)e(the)h(system)g(A)f Fk(X)h Fp(=)g Fk(Bw)p Fp(.)29 b(Maple's)17 b Fk(linsolv)o(e)e Fp(command)f(will)150 2199 y(yield)f(the)i(general)f(solution.)212 2251 y(This)g(general)f (solution)g(can)g(b)q(e)h(put)g(in)o(to)f(the)h(form)d Fk(X)j Fp(=)f Fk(T)h Fp(+)f Fj(\013)g Fk(v1)h Fp(+)f Fj(\014)j Fk(v2)p Fp(,)150 2301 y(where)f(the)g(v)o(ectors)g Fk(v1)f Fp(and)g Fk(v2)g Fp(are)g(in)g(the)g(n)o(ull)f(space)i(of)f(A.)f(These)j(v)o(ectors)f (can)150 2350 y(b)q(e)f(extracted)g(from)d(this)i(general)h(solution)e(b)o(y) g(an)h(adroit)g(use)g(of)g(substitution)g(via)150 2400 y(the)h Fk(subs)f Fp(command.)p eop bop 450 275 a Fp(54)967 b Fm(4.2)13 b(Maple)h(On)g(Line)450 391 y Fe(Exercise)j(2)450 470 y Fp(Ob)r(jection)h(\(b\))g(amoun)o(ts)e(to)i (this:)25 b(If)18 b(the)g(system)f(is)h(inconsisten)o(t,)h(there)g(is)e(no) 450 520 y(solution.)25 b(If)16 b(the)h(system)g(is)f(consisten)o(t,)i(there)f (are)g(man)o(y)e(solutions.)25 b(Ev)o(en)17 b(the)450 570 y(use)g(of)e(pro)r (jections)i(in)f(Exercise)i(1)d(has)i(yielded)f(a)f(general)i(solution)e (that)h(is)g(not)450 620 y(unique.)32 b(P)o(erhaps)19 b(y)o(our)g(\014rst)g (though)o(t)f(w)o(as)g(to)h(rep)q(ort)g(the)g(translation)f(v)o(ector)450 670 y(as)g(the)h(solution.)31 b(But)18 b(there)i(is)e(nothing)g(sp)q(ecial)g (ab)q(out)g(this)h(v)o(ector,)g(and)f(the)450 719 y(T)m(ranslation)13 b(Theorem)g(sa)o(ys)i(the)f(general)h(solution)e(can)h(b)q(e)h(expressed)i
(using)c(an)o(y)450 769 y(particular)h(solution,)e(not)i(just)g(the)h (translation)e(v)o(ector.)512 820 y(Y)m(our)j(next)i(idea,)e(ho)o(w)o(ev)o (er,)h(is)f(sound.)27 b(With)15 b Fk(T)i Fp(as)g(the)g(translation)f(v)o (ector,)450 870 y(let)f Fk(Tn)f Fp(b)q(e)h(its)f(pro)r(jection)h(on)o(to)f (the)h(n)o(ull)f(space)h(of)f(A.)g(Rep)q(ort)h(to)f(y)o(our)g(b)q(oss)h(the) 450 920 y(solution)e Fk(X)h Fp(=)g Fk(T)g Fp(-)g Fk(Tn)p Fp(.)k(Wh)o(y)13 b(is)h(this)g Fk(X)f Fp(a)h(solution?)512 971 y(Use)k(Maple's)e Fk(n)o(ullsp)o(ace)e Fp(command)f(to)k(\014nd)f(a)g(basis)h(for)f(the)h(n)o (ull)e(space)j(of)450 1021 y(A.)d(Then)h(use)g(the)g(F)m(ourier)f(Theorem)g (to)g(obtain)g(the)h(pro)r(jection)g(of)e Fk(T)i Fp(on)o(to)f(this)450 1071 y(n)o(ull)e(space.)20 b(Finally)m(,)12 b(form)g Fk(T)i Fp(-)g Fk(Tn)g Fp(and)g(explain)f(wh)o(y)h(this)g(is)g(a)g(solution.)k(Note,) 450 1120 y(ho)o(w)o(ev)o(er)i(that)h(the)f(F)m(ourier)g(Theorem)g(assumes)g (an)g(orthogonal)f(basis)h(for)f(the)450 1170 y(space)i(in)o(to)f(whic)o(h)g (the)g(pro)r(jection)h(o)q(ccurs.)38 b(Hence,)23 b(y)o(ou)d(will)f(need)i(to) f(apply)450 1220 y(Maple's)13 b Fk(GramSc)o(hmidt)d Fp(command)h(to)i(the)h (basis)g(for)f(the)i(n)o(ullspace.)j(Remem-)450 1270 y(b)q(er,)j(though,)e (that)g Fk(GramSc)o(hmidt)c Fp(returns)21 b(a)e(list)f(of)g(lists,)i (requiring)f(us)g(to)450 1320 y(con)o(v)o(ert)c(the)f(sub-lists)g(bac)o(k)g (to)g(v)o(ectors)h(an)e(w)o(as)h(done)g(in)g(Exercise)h(1.)450 1473 y Fe(Exercise)i(3)450 1552 y Fp(T)m(ry)e(computing)f Fk(X)i Fp(in)f(Exercise)i(2)e(b)o(y)g(starting)h(with)f(a)g(solution)g(other)h(than) f Fk(T)p Fp(.)450 1602 y(Y)m(ou)g(should)h(get)h(the)f(same)f Fk(X)p Fp(.)h(Wh)o(y?)24 b(It)16 b(can)g(b)q(e)h(sho)o(wn)f(that)g(the)h Fk(X)f Fp(found)f(in)450 1652 y(Exercise)h(2)d(is)h(the)g(solution)f(of)h (minim)o(al)c(length.)450 1805 y Fe(Exercise)17 b(4)450 1884 y Fp(Sho)o(w)h(that)h(of)g(all)e(solutions)h(to)h(A)g Fk(X)g Fp(=)g Fk(Bw)p Fp(,)h(the)f(one)g(with)g(minim)o(al)c(length)450 1934 y(is)h(the)g(solution)f Fk(X)g Fp(computed)g(in)h(Exercise)h(3.)23 b(F)m(or)15 b(this,)h(let)g Fk(q1)f Fp(b)q(e)h(the)h(general)450 1984 y(solution)9 b(found)g(b)o(y)h(Maple)g(in)f(Exercise)i(1,)f(considered)i (as)e(a)f(t)o(w)o(o-parameter)g(family)450 2034 y(of)14 b(v)o(ectors.)23 b(Compute)14 b(the)h(2-norm,)e(and)i(use)h(calculus)f(to)g(minim)o(ize)e (this)i(norm.)450 2084 y(The)c(resulting)g(solution)f(should)h(b)q(e)g(the)h (same)e Fk(X)h Fp(that)g(w)o(as)g(computed)f(in)g(Exercise)450 2134 y(3.)512 2185 y(Hin)o(t:)23 b(Use)17 b(the)g Fk(subs)e Fp(command)f(to)i(repace)i(the)f(free)g(parameters)p 1644 2185 13 2 v 32 w(t[1])e(and)p 450 2234 V 462 2234 a(t[2])g(used)h(b)o(y)g(Maple)f (with)g Fl(a)h Fp(and)g Fl(b,)f Fp(calling)g(the)h(result)g Fk(Xg)p Fp(.)23 b(No)o(w,)15 b(obtain)g(the)450 2284 y(2-norm)h(of)h(the)h(v) o(ector)g Fk(Xg)p Fp(.)29 b(Since)18 b Fk(Xg)f Fp(is)h(a)f(sym)o(b)q(olic)e (v)o(ector,)k(Maple)e(returns)450 2334 y(the)f(norm)f(with)g(absolute)h(v)n (alues,)f(thereb)o(y)i(making)c(it)j(di\016cult)f(to)g(di\013eren)o(tiate)450 2384 y(and)j(set)h(deriv)n(ativ)o(es)g(equal)f(to)g(zero.)32 b(Simplify)15 b(the)k(norm)e(of)h Fk(Xg)p Fp(,)h(adding)e(the)450 2434 y(parameter)d Fl(symb)n(olic)f Fp(to)h(coax)g(Maple)f(to)h(simplify)d (the)k(absolute)e(v)n(alues.)p eop bop 150 275 a Fm(4.3)13 b(Maple)h(On)g(Line)967 b(55)212 387 y Fn(>)282 391 y Fo(f)21 b(:=)h(simplify\(norm\(Xg)o(,2\),s)o(ymbol)o(ic\);) 212 470 y Fp(Di\013eren)o(tiate)16 b(with)f(resp)q(ect)j(to)d Fl(a)h Fp(and)f Fl(b.)23 b Fp(Use)16 b(Maple's)f Fk(solv)o(e)f Fp(command)f(to)150 519 y(solv)o(e)e(the)i(system)e(obtained)h(b)o(y)f (setting)h(these)h(partials)e(equal)g(to)h(zero,)g(as)g(follo)o(ws.)150 569 y(Finally)m(,)f(substitute)16 b(these)f(v)n(alues)e(in)o(to)h Fk(Xg)f Fp(and)h(cmpare)g(with)f Fk(X)p Fp(.)212 636 y Fn(>)282 640 y Fo(fa)21 b(:=)h(diff\(f,a\);)d(fb)j(:=)f(diff\(f,b\);)212 706 y Fn(>)282 710 y Fo(qq)g(:=)h(solve\()p Fg(f)p Fo(fa,fb)p Fg(g)p Fo(,)p Fg(f)o Fo(a,b)p Fg(g)o Fo(\);)150 857 y Fe(Exercise)17 b(5)150 934 y Fp(Maple)f(w)o(ould)e(ha)o(v)o(e)i(found)f(the)i(least)f
(squares)h(solution)d(of)i(minim)o(al)c(norm)i(with)150 984 y(its)f(built-in)f Fk(leastsqrs)f Fp(command.)k(V)m(erify)d(that)h(this)g (command)d(yields)j(the)g(solu-)150 1034 y(tion)i(X)h(found)f(in)h(Exercise)h (2.)24 b(Note)16 b(the)g(inclusion)f(of)g(the)i(parameter)e Fl(optimize)150 1083 y Fp(whic)o(h)h(signals)f(Maple)g(to)g(\014nd)h(the)h (solution)d(of)h(minima)o(l)e(length.)23 b(Without)15 b(this)150 1133 y(parameter,)f(the)h Fk(leastsqrs)e Fp(command)f(will)h(return)j(the)g (general)f(solution)e(found)150 1183 y(in)g(Exercise)j(1)e(as)f(the)i(v)o (ector)g(in)e(q1.)212 1249 y Fn(>)282 1253 y Fo(leastsqrs\(A,b,op)o(timiz)o (e\);)212 1320 y Fn(>)282 1324 y Fo(leastsqrs\(A,b\);)150 1470 y Fe(Exercise)k(6)150 1548 y Fp(After)12 b(giving)d(the)j(b)q(oss)f(y)o(our)g (answ)o(er,)h(y)o(ou)f(delete)h(all)d(y)o(our)i(data)g(except)h(for)f(A)g (and)150 1598 y Fk(X)p Fp(.)h(A)h(mon)o(th)d(later)j(the)g(customer)g(calls,) e(sa)o(ying,)h("W)m(e)g(kno)o(w)g(that)g(there)i(m)o(ust)d(b)q(e)150 1647 y(other)k(solutions.)i(Could)c(y)o(ou)h(please)g(pro)o(vide)g(us)g(with) g(the)g(general)g(solution?")212 1698 y(Sho)o(w)e(ho)o(w)f(the)h(general)g (solution)f(can)h(b)q(e)h(reconstructed)h(from)c(A)i(and)g Fk(X)f Fp(with)150 1747 y(a)j(single)f(Maple)h(command.)150 1943 y Fq(4.3)66 b(On)22 b(Line)150 2035 y Fp(Restart)11 b(Maple)g(to)f (clear)h(its)g(memory)d(of)i(all)g(previously)g(de\014ned)i(v)n(ariables.)k (Then,)150 2085 y(re-initialize)c(b)o(y)h(loading)e(the)j Fl(linalg)f Fp(and)g Fl(plots)g Fp(pac)o(k)n(ages.)k(In)c(addition,)f(en)o(ter)i(the)150 2134 y(follo)o(wing)g(lines)j(of)f(Maple)g(co)q(de.)27 b(This)17 b(co)q(de,)h(written)f(b)o(y)f(Dr.)26 b(Mik)o(e)17 b(Monagan)150 2184 y(of)12 b(Simon)e(F)m(rasier)j(Univ)o(ersit)o(y)f(in)h(Burnab)o(y)m(,)f (British)g(Colom)o(bia,)d(Canada,)j(creates)150 2234 y(a)19 b(function)g(that)h(will)e(generate)i(the)g(p)q(erio)q(dic)g(extension)g(of)f (a)g(function.)35 b(The)150 2284 y(co)q(de)17 b(\014rst)f(app)q(eared)h(in)e (the)h(article)g(Tips)e(for)f(Maple)h(Users)h(and)f(Programmers)p 768 2299 721 2 v -2 w(,)150 2334 y Fl(MapleT)m(e)n(ch)p Fp(,)g(V)o(OL.)f(3,)g (NO.)h(3,)f(1996,)g(published)h(b)o(y)f Fl(Birkhauser)p Fp(.)212 2384 y(PE)f(:=)f(pro)q(c\(f,)g(d::range\))g(subs\()p Fg(f)p Fp('F')g(=)g(f,)g('L')f(=)h(lhs\(d\),)g('D')f(=)i(rhs\(d\)-lhs\(d\))p Fg(g)p Fp(,)150 2434 y(pro)q(c\(x::algebraic\))h(lo)q(cal)g(y;)h(y)f(:=)h (\015o)q(or\(\(x-L\)/D\);)f(F\(x-y*D\);)g(end\))i(end:)p eop bop 450 275 a Fp(56)967 b Fm(4.3)13 b(Maple)h(On)g(Line)512 391 y Fp(These)f(lines)e(of)g(co)q(de)h(can)f(b)q(e)h(en)o(tered)h(in)o(to)e (a)g(separate)h(Maple)g(w)o(orksheet,)g(and)450 441 y(that)17 b(w)o(orksheet)h(sa)o(v)o(ed.)26 b(Later,)17 b(if)f(the)h(co)q(de)h(is)e (again)g(needed,)i(that)f(w)o(orksheet)450 491 y(can)e(b)q(e)g(op)q(ened,)g (and)g(the)g(lines)f(copied)h(and)f(pasted)i(in)o(to)d(the)j(activ)o(e)e(w)o (orksheet.)450 541 y(There)k(are,)g(of)f(course,)i(other)f(w)o(a)o(ys)f(of)f (sa)o(ving)h(co)q(de)h(and)f(making)e(it)i(accessible)450 591 y(more)d(easily)m(,)f(but)j(some)d(asp)q(ects)k(of)d(that)h(pro)q(cess)i(are) e(platform)e(dep)q(enden)o(t)j(and)450 640 y(will)c(not)i(b)q(e)h(discussed)g (here.)450 788 y Fe(Exercise)i(1)450 866 y Fp(Figure)h(1)f(of)h(Section)g (6.6)e(in)i(the)g(text)g(depicts)h(a)f(sa)o(w-to)q(oth)f(function)h(called)f (a)450 916 y("rasp.")12 b(Plot)f(the)i(\014rst,)g(fourth,)f(and)g(ten)o(th)h (F)m(ourier)f(sine)g(appro)o(ximations)e(to)i(this)450 966 y(function.)512 1016 y(The)k(rasp)f(is)g(generated)i(b)o(y)e(the)g(p)q(erio)q (dic)h(extension)g(of)e(the)i(function)f(f\(x\))g(=)450 1066 y(x,)d(for)f(x)h(in)g(the)h(in)o(terv)n(al)e([-1,1].)k(T)m(o)c(get)i(Maple)e (to)h(plot)g(the)g(p)q(erio)q(dic)h(extension)g(of)450 1116 y(f\(x\),)g(use)i(the)g(function)e(PE)i(de\014ned)g(in)e(the)i(In)o(tro)q (duction)f(ab)q(o)o(v)o(e.)k(First,)13 b(b)q(e)i(sure)450 1166 y(to)i(de\014ne)h(f\(x\))g(with)f(Maple's)g(arro)o(w)g(notation,)g(thereb)o (y)h(making)d(f)i(a)g(function,)450 1216 y(not)d(an)f(expression.)512 1283 y Fn(>)582 1287 y Fo(f)21 b(:=)h(x)f(->)h(x;)512 1366 y Fp(De\014ne)g(the)f(function)f(whose)h(name)f(is)g("rasp")g(as)h(the)g(p)q
(erio)q(dic)g(extension)450 1415 y(\(hence,)16 b(PE\))f(of)f(the)h(function)g (whose)g(name)e(is)i(f.)k(Do)c(this)f(b)o(y)h(applying)e(the)i Fk(PE)450 1465 y Fp(op)q(erator)g(to)f(f,)g(b)q(eing)g(sure)h(to)f(terminate) g(the)h(command)c(with)j(a)g(colon)g(\(:\))19 b(since)450 1515 y(the)e(output)f(will)e(lo)q(ok)i(strange,)g(and)g(probably)g(unin)o (telligible.)22 b(The)17 b(argumen)o(ts)450 1565 y(to)h Fk(PE)g Fp(are)g(the)h(function)f(to)g(b)q(e)h(extended,)h(and)e(the)g(domain)e(of)i (the)g(function)450 1615 y(b)q(eing)p 450 1629 99 2 v 14 w(extended.)512 1689 y Fn(>)582 1693 y Fo(rasp)j(:=)g(PE\(f,-1..1\):)512 1772 y Fp(Obtain)15 b(a)g(graph)h(of)e(the)i(rasp)g(on)f(the)h(in)o(terv)n(al)f ([-3,3],)e(assigning)h(the)i(plot)f(to)450 1822 y(a)h(v)n(ariable)g(for)g (use)i(later.)26 b(The)17 b(plot)f(option)g Fl(disc)n(ont)h(=)h(true)e Fp(signals)g(Maple)g(to)450 1871 y(observ)o(e)c(the)g(discon)o(tin)o(uities)f (in)g(the)g(function,)g(and)g(tells)h(it)e(not)h(to)h(connect)g(across)450 1921 y(the)i(jumps.)512 1988 y Fn(>)582 1992 y Fo(f1:=plot\(rasp\(x\))o (,x=-3)o(..3,)k(discont=true,)i(color=black,)f(scaling=constrain)o(ed,)512 2035 y(thickness=3\):)41 b(f1;)512 2114 y Fp(Obtain)13 b(the)h(F)m(ourier)e (sine)i(series)g(co)q(e\016cien)o(ts)g Fj(b)1285 2120 y Fn(n)1321 2114 y Fp(=)f([)1383 2097 y Fi(2)p 1382 2104 17 2 v 1385 2128 a Fn(l)1404 2114 y Fp(])1429 2080 y Ff(R)1457 2090 y Fn(l)1449 2128 y Fi(0)1476 2114 y Fj(f)t Fp(\()p Fj(x)p Fp(\))7 b Fj(sin)r Fp(\()1644 2097 y Fn(n)e(\031)i(x)p 1644 2104 72 2 v 1674 2128 a(l)1720 2114 y Fp(\))g Fj(dx)p Fp(.)450 2163 y(Here,)21 b Fl(l)d Fp(=)i(1)e(and)h(f\(x\))g(=)g(x.)33 b(An)19 b(in)o(tegral)f(can)h(b)q (e)h(en)o(tered)g(in)o(to)f(Maple)f(with)450 2213 y(the)f Fk(In)o(t)d Fp(command)g(whic)o(h)i(stores)h(the)g(in)o(tegral)e(as)h(an)g(unev)n (aluated)g(sym)o(b)q(ol.)22 b(If)450 2263 y(instead,)c(the)h(in)o(tegral)d (is)i(en)o(tered)h(with)e(the)h Fk(in)o(t)e Fp(command,)f(the)j(ev)n (aluation)f(of)450 2313 y(the)h(in)o(tegral)g(is)f(immediate.)27 b(Here,)20 b Fk(In)o(t)c Fp(is)i(used)h(so)f(that)f(the)i(in)o(tegral)e(will) f(b)q(e)450 2363 y(displa)o(y)o(ed)d(completely)m(.)512 2430 y Fn(>)582 2434 y Fo(q)21 b(:=)h(2*Int\(x*sin\(n*Pi)o(*x\),x)o(=0..1)o(\);)p eop bop 150 275 a Fm(4.3)13 b(Maple)h(On)g(Line)967 b(57)212 391 y Fp(T)m(o)14 b(ev)n(aluate)h(an)g(in)o(tegral)f(that)h(has)g(b)q(een)h(en)o (tered)g(with)f Fk(In)o(t)p Fp(,)e(apply)h(Maple's)150 441 y Fk(v)m(alue)f Fp(command)e(to)j(the)h(in)o(tegral.)212 523 y Fn(>)282 527 y Fo(q1)21 b(:=)h(value\(q\);)212 625 y Fp(W)m(e)11 b(wish)f(to)h(simplify)d(this)j(expression.)18 b(F)m(or)10 b(example,)g Fj(sin)p Fp(\()p Fj(n)d(\031)q Fp(\))k(is)g(zero)h(when-)150 675 y(ev)o(er)h Fl(n)g Fp(is)g(an)f(in)o(teger.)18 b(W)m(e)12 b(will)f(tell)h(Maple)h(that)f Fl(n)h Fp(is)f(an)h(in)o(teger)g(with)f(its)g (assume)150 725 y(command.)19 b(Ho)o(w)o(ev)o(er,)c(that)g(will)f(cause)i (Maple)f(to)g(attac)o(h)g(a)g(tilde)g(\(~\))g(to)g(eac)o(h)h Fl(n)150 774 y Fp(it)f(prin)o(ts)g(thereafter.)23 b(Suppressing)16 b(the)g(attac)o(hmen)o(t)e(of)h(the)h(tildes)f(can)g(b)q(e)h(done)150 824 y(either)h(in)o(teractiv)o(ely)e(from)g(the)h(Options)g(men)o(u)f (\(Options,)h(Assumed)g(V)m(ariables,)150 874 y(No)i(Annotation\))f(or)h (from)e(the)i(command)d(line)i(with)h(the)g(follo)o(wing)e Fk(in)o(terface)150 924 y Fp(command.)212 1005 y Fn(>)282 1009 y Fo(interface\(showas)o(sumed)o(=0\);)212 1108 y Fp(No)o(w,)d(use)i(the)g Fk(assume)d Fp(command)g(to)h(tell)h(Maple)f(that)h Fl(n)h Fp(is)e(an)h(in)o(teger.)212 1189 y Fn(>)282 1193 y Fo(assume\(n,integer)o (\);)212 1291 y Fp(If)e(the)h(F)m(ourier)g(co)q(e\016cien)o(ts)g(are)g(no)o (w)f(simpli\014ed,)f(they)i(will)e(app)q(ear)h(m)o(uc)o(h)g(lik)o(e)150 1341 y(they)f(w)o(ould)e(if)h(the)g(calculation)g(w)o(ere)h(done)f("b)o(y)g (hand.")f(Note)i(ho)o(w,)f(in)g(the)h(in)o(terest)150 1391 y(of)16 b(simplicit)o(y)m(,)d(w)o(e)k(assign)f(the)h(result)g(to)f(the)h (name)f("b")g(and)g(not)g(to)g(something)150 1441 y(that)e(tries)h(to)e
(re\015ect)j(the)e(dep)q(endence)j(on)d Fl(n)p Fp(.)212 1522 y Fn(>)282 1526 y Fo(b)21 b(:=)h(simplify\(q1\);)212 1625 y Fp(The)16 b(F)m(ourier)f(appro)o(ximations)e(are)i(simply)f(partial)g(sums)h (of)f(the)i(F)m(ourier)f(se-)150 1674 y(ries.)26 b(The)17 b(\014rst)g(appro)o (ximation,)c(p1,)j(is)h(just)f Fj(b)936 1680 y Fi(1)961 1674 y Fj(sin)q Fp(\()p Fj(\031)8 b(x)p Fp(\))17 b(and)f(the)h(ten)o(th)g(one)f (is)150 1729 y Fj(p)p Fp(10)g(=)277 1697 y Ff(P)321 1708 y Fi(10)321 1741 y Fn(n)p Fi(=1)399 1729 y Fj(b)417 1735 y Fn(n)447 1729 y Fj(sin)p Fp(\()p Fj(n)7 b(\031)h(x)p Fp(\).)26 b(W)m(e)17 b(can)f(obtain)g(these)i(expressions)h(in)d(Maple)g(b)o(y)150 1778 y(using)e(its)g Fk(sum)e Fp(command.)212 1860 y Fn(>)282 1864 y Fo(p1:=sum\(b*sin\(n*)o(Pi*x\))o(,n=1.)o(.1\);)18 b (p4:=sum\(b*sin\(n*Pi*)o(x\),n=)o(1..4\))o(;)212 1906 y (p10:=sum\(b*sin\(n*Pi)o(*x\),n)o(=1..)o(10\);)212 2004 y Fp(Finally)m(,)10 b(these)k(three)g(partial)e(sums)g(can)g(b)q(e)i(plotted)e(with)g(a)h(single) f Fk(plot)e Fp(com-)150 2054 y(mand)16 b(b)o(y)h(grouping)f(the)i(functions)f (in)g(a)g(list.)28 b(Colors)16 b(can)i(b)q(e)g(assigned)f(to)g(the)150 2104 y(functions)g(b)o(y)f(the)h(option)e Fl(c)n(olor)h Fp(=)h([...],)d(with) i(matc)o(hing)f(colors)i(b)q(eing)f(listed)g(in)150 2154 y(the)f(order)g(of)e (the)i(functions)f(to)g(whic)o(h)g(they)g(are)h(b)q(eing)f(ascrib)q(ed.)20 b(Assigning)13 b(the)150 2204 y(plot)j(to)h(a)g(v)n(ariable)f(allo)o(ws)g (merging,)g(via)g(the)h Fk(displa)o(y)e Fp(command)f(of)j(the)g Fl(plots)150 2254 y Fp(pac)o(k)n(age,)c(the)i(appro)o(ximatio)o(ns)d(with)h (the)i(graph)f(of)f(the)h(rasp)h(created)g(ab)q(o)o(v)o(e)212 2335 y Fn(>)282 2339 y Fo(f2)21 b(:=)h(plot\([p1,p4,p10)o(],x=-)o(3..3,)o (color)o(=[red)o(,)d(green,)h(blue]\):)212 2381 y(display\([f1,f2]\);)p eop bop 450 275 a Fp(58)967 b Fm(4.3)13 b(Maple)h(On)g(Line)450 391 y Fe(Exercise)j(2)450 475 y Fp(Let)c Fj(g)q Fp(\()p Fj(x)p Fp(\))f(=)656 429 y Ff(n)691 450 y Fg(\000)p Fp(1)41 b Fj(x)11 b(<)h Fp(0)691 500 y(1)73 b(0)12 b Fg(\024)f Fj(x)890 475 y Fp(,)i(a)f(piecewise)j(de\014ned)f(function.)j(Obtain)c(F)m(ourier)g(sine)450 538 y(appro)o(ximations)e(with)j(4,)f(8,)g(and)h(20)f(sine)h(functions.)512 589 y(The)f(p)q(erio)q(dic)g(extension)g(of)f(g\(x\))g(is)g(called)h(a)f ("square)h(w)o(a)o(v)o(e.")e(Note)i(the)g("ear-)450 639 y(lik)o(e")d(p)q (eaks)i(whic)o(h)f(app)q(ear)h(in)f(the)h(graph)f(of)g(the)h(partial)e(sums)g (at)i(the)f(discon)o(tin)o(u-)450 688 y(ities)h(of)f(f\(x\).)17 b(These)c(p)q(eaks)g(are)f(referred)i(to)e(as)g(the)g("Gibbs)f(phenomenon.")g (They)450 738 y(are)k(quite)f(pronounced,)i(ev)o(en)f(after)f(t)o(w)o(en)o(t) o(y)h(terms)f(of)g(the)h(F)m(ourier)g(series.)21 b(Their)450 788 y(existence)d(sho)o(ws)e(that)f(it)h(tak)o(es)g(a)g(v)o(ery)g(high)f (\014delit)o(y)g(ampli\014er)f(to)i(repro)q(duce)h(a)450 838 y(square)e(w)o(a)o(v)o(e)g(accurately)m(.)20 b(F)m(or)14 b(this)h(reason,)g (square)h(w)o(a)o(v)o(es)e(are)h(sometimes)e(used)450 888 y(to)h(test)h(the)f (\014delit)o(y)g(of)f(an)g(ampli\014er.)512 938 y(Begin)f(b)o(y)f(de\014ning) h(g\(x\))f(as)h(a)f(piecewise)i(function)f(on)f(the)h(in)o(terv)n(al)f ([-1,1].)k(Use)450 988 y(Maple's)h Fk(piecewise)f Fp(function)h(whic)o(h)g(p) q(ermits)g(the)h(de\014nition)f(a)g(function)g(with)450 1038 y(m)o(ultiple)c(form)o(ulas.)j(De\014ne)g(g)f(as)g(a)f(function)h(b)o(y)f (using)h(the)g(arro)o(w)g(notation.)512 1104 y Fn(>)582 1108 y Fo(g)21 b(:=)h(x)f(->)h(piecewise\(x<0,-1)o(,x>=0)o(,1\);)512 1187 y Fp(Chec)o(k)17 b(the)g(b)q(eha)o(vior)f(of)f(g\(x\))h(b)o(y)g (plotting)f(it,)h(again)f(using)g(the)i(plot)f(option)450 1237 y Fl(disc)n(ont)f(=)g(true)e Fp(so)h(that)g(jumps)f(in)g(the)i(function)e (are)i(not)e(connected.)512 1303 y Fn(>)582 1307 y Fo(plot\(g\(x\),x=-1..)o (1,dis)o(cont=)o(true,)18 b(color)j(=)h(black\);)512 1386 y Fp(De\014ne)16 b(G)e(as)h(the)g(p)q(erio)q(dic)g(extension)g(of)f(the)i (function)e(g.)20 b(Use)c(the)f(PE)g(co)q(de)450 1436 y(detailed)d(in)f(the)h (In)o(tro)q(duction.)18 b(Again,)11 b(end)h(the)h(command)8
b(with)k(a)f(clon)h(\(:\))17 b(since)450 1486 y(the)c(ec)o(ho)f(will)f (probably)g(not)h(mak)o(e)f(m)o(uc)o(h)g(sense)j(to)e(the)g(t)o(ypical)g (studen)o(t)h(of)e(linear)450 1535 y(algebra.)24 b(Plot)15 b(G)h(on)f(the)i(in)o(terv)n(al)e([-3,3],)f(assigning)h(the)i(plot)e(to)h(a)f (v)n(ariable)g(for)450 1585 y(use)g(later.)512 1635 y(As)g(in)f(Exercise)i (1,)d(the)i(F)m(ourier)g(sine)f(co)q(e\016cien)o(ts)i Fj(b)1362 1641 y Fn(n)1398 1635 y Fp(are)f(computed)f(b)o(y)g(in)o(te-)450 1685 y(grating)e(g\(x\).)17 b(En)o(ter)d(the)f(de\014ning)g(in)o(tegral)f (using)g Fk(in)o(t)f Fp(and)i(g\(x\).)k(The)c(presump-)450 1735 y(tion)h(here)i(is)f(that)g(the)h Fk(in)o(terface)c Fp(and)j Fk(assume)f Fp(commands)e(are)k(still)e(op)q(erativ)o(e)450 1785 y(from)e(Exercise)k(1.)h(If)d(not,)f(re-execute)k(those)d(commands.)450 1932 y Fe(Exercise)j(3)450 2010 y Fp(Obtain)d(the)i(thirtieth)f(partial)e (sum)h(of)g(the)h(F)m(ourier)g(sine)g(series)h(for)e(the)h(function)450 2059 y Fj(f)t Fp(\()p Fj(x)p Fp(\))d(=)621 2043 y Fi(1)p 591 2050 78 2 v 591 2074 a(1+)p Fn(x)652 2066 y Fd(2)673 2059 y Fp(,)f Fg(\000)p Fp(1)h Fg(\024)f Fj(x)g Fg(\024)18 b Fp(1,)11 b(then)g(graph)g(the)g(p)q(erio)q(dic)g(extension)g(of)f(f\(x\))h(and)f(the) 450 2109 y(F)m(ourier)h(appro)o(ximation.)j(Explain)c(wh)o(y)g(the)i(graphs)f (don't)f(agree.)18 b(\(See)12 b(Exercises)450 2159 y(6)i(and)f(7)h(of)f (Section)h(6.6\).)450 2306 y Fe(Exercise)j(4)450 2384 y Fp(Obtain)c(a)g(F)m (ourier)g(cosine)h(series)g(for)f(the)h(function)f(in)g(Exercise)h(3.)k(Plot) 13 b(the)h(\014rst,)450 2434 y(fourth,)f(and)h(eigh)o(th)g(partial)f(sums.)p eop bop 150 275 a Fm(4.4)13 b(Maple)h(On)g(Line)967 b(59)150 391 y Fq(4.4)66 b(On)22 b(Line)150 492 y Fp(Restart)14 b(Maple)f(to)g(clear)g (its)g(memory)e(of)i(all)f(de\014ned)i(v)n(ariables,)e(and)h(re-initialize) 150 542 y(b)o(y)h(loading)e(the)i Fl(linalg)g Fp(and)g Fl(plots)f Fp(pac)o(k)n(ages.)212 597 y(Let)20 b Fj(R)324 603 y Fn(x)344 597 y Fp(\()365 581 y Fn(\031)p 365 588 21 2 v 367 611 a Fi(6)391 597 y Fp(\))f(b)q(e)h(the)g(matrix)d(of)i(a)f(coun)o(terclo)q(c)o(kwise)j (rotation,)e(around)g(the)150 647 y Fl(x)p Fp(-axis)g(and)f(through)h(an)g (angle)g(of)754 630 y Fn(\031)p 754 637 V 756 661 a Fi(6)798 647 y Fp(radians.)34 b(Let)19 b Fj(R)1087 653 y Fn(y)1107 647 y Fp(\()1128 630 y Fn(\031)p 1128 637 V 1130 661 a Fi(4)1153 647 y Fp(\))g(b)q(e)h(the)f(matrix)f(of)150 697 y(a)h(coun)o(terclo)q(c)o (kwise)j(rotation,)e(around)g(the)g Fl(y)p Fp(-axis)f(and)h(through)g(an)f (angle)h(of)155 730 y Fn(\031)p 155 737 V 157 761 a Fi(4)198 746 y Fp(radians.)29 b(Let)18 b(A)g(=)g Fj(R)580 752 y Fn(x)601 746 y Fp(\()622 730 y Fn(\031)p 622 737 V 624 761 a Fi(6)647 746 y Fp(\))7 b Fj(R)702 752 y Fn(y)722 746 y Fp(\()743 730 y Fn(\031)p 743 737 V 745 761 a Fi(4)768 746 y Fp(\).)30 b(Since)18 b(the)g(pro)q(duct)h(of)e(t)o(w)o(o)g(orthogonal)150 796 y(matrices)e(is)g (orthogonal,)f(A)h(is)h(orthogonal.)21 b(The)16 b(purp)q(ose)g(of)f(these)i (exercises)g(is)150 846 y(a)e(demonstration)f(that)h(A)g(de\014nes)h(a)f (rotation)f(ab)q(out)h(a)g(\014xed)g(axis)g(and)g(through)150 896 y(a)f(particular)f(angle.)18 b(See)d(Figure)f(5)f(in)h(Section)g(4.4)f (of)g(the)i(text..)212 951 y(P)o(oin)o(ts)i(on)f(this)h(axis)f(remain)g (\014xed)h(under)g(the)h(rotation.)26 b(Th)o(us,)17 b(if)f Fk(X)h Fp(is)f(on)150 1001 y(the)e(axis)g(of)f(rotation,)f(it)i(will)e (satisfy)h(A)p Fk(X)h Fp(=)g Fk(X)p Fp(,)f(or)h(equiv)n(alen)o(tly)m(,)d(\(A) j(-)g(I\))p Fk(X)g Fp(=)g Fk(0)p Fp(.)150 1188 y Fe(Exercise)j(1)150 1274 y Fp(Construct)e(the)g(matrix)d(A)i(as)g(the)g(pro)q(duct)h(of)e(the)i (matrices)e Fj(R)1191 1280 y Fn(x)1211 1274 y Fp(\()1232 1258 y Fn(\031)p 1232 1265 V 1234 1288 a Fi(6)1258 1274 y Fp(\))h(and)g Fj(R)1401 1280 y Fn(y)1420 1274 y Fp(\()1441 1258 y Fn(\031)p 1441 1265 V 1443 1288 a Fi(4)1467 1274 y Fp(\).)150 1461 y Fe(Exercise)j(2)150 1547 y Fp(Find)h(an)f Fk(X)h Fp(on)g(the)h(axis)e(of)g (rotation)h(b)o(y)g(using)f(the)i(equation)e(\(A)i(-)e(I\))p
Fk(X)h Fp(=)h Fk(0)p Fp(.)150 1597 y(Th)o(us,)14 b Fk(X)f Fp(is)h(in)f(the)i (n)o(ull)e(space)h(of)g(A)g(-)f(I.)g(Suc)o(h)i(an)e Fk(X)h Fp(can)g(b)q(e)g(found)g(b)o(y)f(applying)150 1647 y(Maple's)k Fk(n)o(ullsp)o(ace)e Fp(command)f(to)j(A)g(-)g(I,)g(whic)o(h)g(Maple)g(lets)g (us)h(form)d(via)h(the)150 1697 y(syn)o(tax)e(A)h(-)f(1.)19 b(The)c Fk(n)o(ullsp)o(ace)d Fp(command)g(returns)j(a)f(set)i(of)d(v)o (ectors,)j(so)e Fk(X)g Fp(will)150 1747 y(ha)o(v)o(e)g(to)f(b)q(e)i (extracted)g(from)e(this)g(set.)150 1934 y Fe(Exercise)k(3)150 2020 y Fp(Plot)f(the)i(line)e(segmen)o(t)h(from)e(-)p Fk(X)h Fp(to)h Fk(X)p Fp(.)f(If)h(w)o(e)g(parametrize)f(this)h(line)f(segmen)o(t)150 2070 y(as)h(t)p Fk(X)g Fp(w)o(e)g(can)g(plot)f(it)g(with)h(the)g Fk(spacecurv)o(e)e Fp(command,)g(letting)h(t)h(lie)f(in)g(the)150 2120 y(in)o(terv)n(al)g([-1,1].)26 b(Assign)17 b(this)g(plot)f(to)h(a)g(v)n (ariable)f(so)h(it)g(can)g(b)q(e)h(used)g(in)e(a)h(later)150 2169 y(exercise.)212 2250 y Fn(>)282 2254 y Fo(tX)k(:=)h(evalm\(t*X\);)212 2335 y Fn(>)282 2339 y Fo(f1)f(:=)h(spacecurve\(tX,t)o(=-1..)o(1,)d (color=black,)g(axes=boxed,)h(scaling=constrai)o(ned,)212 2382 y(labels=[x,y,z],)f(labelfont=[TIMES,)o(BOLD,)o(14]\):)40 b(f1;)p eop bop 450 275 a Fp(60)967 b Fm(4.4)13 b(Maple)h(On)g(Line)450 391 y Fe(Exercise)j(4)450 474 y Fp(The)e(plane)f(P)h(through)f(the)h(origin)e (p)q(erp)q(endicular)j(to)e Fk(X)g Fp(is)h(called)f(the)h("plane)f(of)450 523 y(rotation.")i(Since)h(this)g(plane)g(con)o(tains)g(the)h(origin,)e(it)h (is)g(a)f(subspace)j(of)d Fj(R)1714 508 y Fi(3)1733 523 y Fp(.)27 b(If)450 573 y(the)15 b(v)o(ector)g Fk(X)f Fp(is)g(con)o(v)o(erted)i(to)e(a)g (1)g(x)g(3)g(matrix,)e(Maple's)i Fk(n)o(ullsp)o(ace)e Fp(command)450 623 y(will)i(pro)q(duce)j(a)e(basis)h(for)f(the)h(plane)f(P)h(\(the)g(n)o (ull)f(space)i(of)e(the)h(matrix-form)c(of)450 673 y Fk(X)p Fp(\).)j(Wh)o(y?)20 b(Obtain)15 b(this)g(basis,)g(nameing)e(its)i(elemen)o (ts)g Fk(N1)g Fp(and)g Fk(N2)p Fp(.)21 b(Plot)14 b(line)450 723 y(segmen)o(ts)j(through)g Fk(N1)h Fp(and)f Fk(N2)g Fp(as)h(w)o(as)f(done) g(in)g(Exercise)i(3.)27 b(Join)17 b(this)h(plot)450 772 y(with)13 b(the)g(one)h(from)d(from)g(Exercise)k(3)e(with)f Fk(displa)o(y3d)e Fp(from)h(the)j Fl(plots)f Fp(pac)o(k)n(age,)450 822 y(and)h(assign)h(the)g (merged)f(graphs)h(to)g(a)f(v)n(ariable)f(for)i(use)g(later)g(in)f(Exercise)i (4.)k(Be)450 872 y(sure)13 b(to)f(use)g(a)g(1-1)f(asp)q(ect)i(ratio)f(so)g (that)g(orthogonal)e(v)o(ectors)j(app)q(ear)g(orthogonal.)512 925 y(The)j(con)o(v)o(ersion)g(of)f Fk(X)j Fp(to)e(a)f(matrix)f(is)h (accomplished)g(b)o(y)g(Maple's)h Fk(con)o(v)o(ert)450 975 y Fp(command,)f(with)i("matrix")e(as)i(the)h(parameter.)27 b(This)17 b(will)f(b)q(e)i(a)f(3)g(x)g(1)g(matrix)450 1025 y(whic)o(h)c(the)h Fk(n)o(ullspace)c Fp(command)h(will)h(reject.)19 b(Apply)13 b(the)h Fk(transp)q(ose)d Fp(op)q(erator)450 1074 y(to)j(pro)q(duce)h(a)e(1)h(x)g(3)f(matrix)f(to)i(giv)o(e)f(to)h(the)h Fk(n)o(ullsp)o(ace)c Fp(command.)512 1127 y(Maple's)k Fk(n)o(ullsp)o(ace)d Fp(command)h(do)q(es)i(not)g(yield)f(normalized)f(v)o(ectors.)22 b(After)450 1177 y(obtaining)15 b(the)j(basis)f(of)f(the)h(n)o(ull)f(space,)i (normalize)d(the)j(v)o(ectors)g(with)e(Maple's)450 1227 y Fk(normalize)11 b Fp(command.)512 1280 y(Sho)o(w)g(that)g(the)g(basis)g(v)o(ectors)h(are)f (not)f(necessarily)i(orthogonal)e(to)h(eac)o(h)g(other.)450 1329 y(They)h(are)g(orthogonal)e(to)h Fk(X)p Fp(.)g(Use)i(Maple's)e Fk(dotpro)q(d)e Fp(command)g(to)i(compute)g(the)450 1379 y(dot)j(pro)q(ducts) h(of)e(v)o(ectors.)450 1547 y Fe(Exercise)k(5)450 1629 y Fp(The)11 b(exp)q(ectation)h(should)f(b)q(e)g(that)g(m)o(ultiplication)d(b)o(y)i(A)h (rotates)h(elemen)o(ts)f(within)450 1679 y(the)k(plane)g(orthogonal)e(to)h (the)i(axis)e(of)g(rotation.)19 b(T)m(o)14 b(test)i(this)e(h)o(yp)q(othesis,) h(m)o(ul-)450 1729 y(tiply)d(b)q(oth)h Fk(N1)g Fp(and)g Fk(N2)g Fp(b)o(y)g(A.)f(Then)i(use)f(Maple's)g(angle)f(command)f(to)h(\014nd)h(the) 450 1779 y(angle)g(b)q(et)o(w)o(een)j Fk(N1)e Fp(and)f(A)h
Fk(N1)p Fp(,)g(and)g(b)q(et)o(w)o(een)h Fk(N2)f Fp(and)g(A)g Fk(N2)p Fp(.)512 1832 y(A)k(second)h(m)o(ultiplicatio)o(n)c(b)o(y)j(A)f (should)h(rotate)g(A)g Fk(N1)g Fp(and)f(A)h Fk(N2)g Fp(b)o(y)f(the)450 1882 y(same)c(amoun)o(t.)j(V)m(erify)e(this.)450 2050 y Fe(Exercise)j(6)450 2132 y Fp(Con)o(tin)o(ue)h(to)g(explore,)h(b)o(y)e(visual)h(means,)f(the)i (idea)f(of)f(rotations)h(in)g(the)g(plane)450 2182 y(P)m(.)c(Create)i(a)e (plot)g(of)g(15)g(successiv)o(e)j(applications)d(of)g(the)h(matrix)e(A)i(to)g (the)g(basis)450 2231 y(v)o(ectors)k(found)e(in)h(Exercise)h(2.)30 b(If)17 b(eac)o(h)h(application)f(of)g(A)h(rotates)h(these)g(basis)450 2281 y(v)o(ectors)j(through)e(a)h(\014xed)g(angle,)g(and)g(if)e(they)j (remain)d(in)h(P)m(,)g(their)h(collectiv)o(e)450 2331 y(image)12 b(should)i("sho)o(w")f(the)h(plane)g(P)m(.)512 2384 y(Since)19 b(w)o(orking)e(with)h(exact)h(expressions)h(can)f(lead)f(to)g(memory-consumi) o(ng)450 2434 y("expression)f(sw)o(ell,")f(it)g(is)h(wise)f(to)h(con)o(v)o (ert)g(the)g(computations)e(to)i(the)g(n)o(umeric)p eop bop 150 275 a Fm(4.5)13 b(Maple)h(On)g(Line)967 b(61)150 391 y Fp(domain.)24 b(Map)17 b(the)g Fk(con)o(v)o(ert)e Fp(op)q(erator,)j(with)e (the)i Fl(\015o)n(at)f Fp(option,)f(on)o(to)g(the)i(ma-)150 441 y(trix)d(A)h(and)f(the)h(v)o(ectors)h Fk(N1)e Fp(and)g Fk(N2)p Fp(,)h(coining)e(new)i(names)f(for)g(these)i(n)o(umeric)150 491 y(v)o(ersions.)29 b(F)m(or)17 b(example,)f(w)o(e)h(migh)o(t)f(call)g(the) i(\015oating)e(p)q(oin)o(t)h(v)o(ersions)h(of)f(these)150 541 y(quan)o(tities)d(B,)f Fk(NN1)p Fp(,)h(and)g Fk(NN2,)h Fp(resp)q(ectiv)o(ely) m(.)212 594 y(Multiplication)e(of)h Fk(NN1)h Fp(b)o(y)f(B,)h(k-times,)e(pro)q (duces)j Fj(B)1126 579 y Fn(k)1147 594 y Fk(NN1)p Fp(.)k(W)m(e)14 b(can)h(pro-)150 643 y(duce)g(the)g(desired)h(plots)e(using)g(Maple's)g(seq)h (command)d(as)i(follo)o(ws.)k(Finally)m(,)12 b(use)150 693 y Fk(displa)o(y3d)f Fp(to)i(merge)h(these)h(plots)f(with)f(the)i(plot)e(from) f(Exercise)k(4.)212 768 y Fn(>)282 772 y Fo(s1)21 b(:=)h(seq\(evalm\(t*\(\(B) o(^k\))d(&*)i(NN1\)\),k=1..15\):)40 b(s2)22 b(:=)f(seq\(evalm\(t*\(\(B^k\)) 212 814 y(&*)h(NN2\)\),k=1..15\):)212 888 y Fn(>)282 892 y Fo(f4)f(:=)h(spacecurve\()p Fg(f)p Fo(s1)p Fg(g)o Fo(,t=0.)o(.1,co)o(lor=)o (blue\))o(:)41 b(f5)21 b(:=)h(spacecurve\()p Fg(f)p Fo(s2)p Fg(g)o Fo(,t=0.)o(.1,co)o(lor=g)o(reen)o(\):)150 1060 y Fe(Exercise)17 b(7)150 1143 y Fp(Determine,)f(in)f(the)i(plane)f(P)m(,)f(the)i(angle)f(of)f (rotation)g(caused)j(b)o(y)d(m)o(ultiplication)150 1193 y(b)o(y)f(A.)212 1245 y(Use)k(Maple's)f(angle)g(command,)e(but)i(express)i(the)f(answ)o(er)g (in)f(degrees,)i(as)e(a)150 1295 y(\015oating)c(p)q(oin)o(t)g(n)o(um)o(b)q (er.)150 1521 y Fq(4.5)66 b(On)22 b(Line)150 1617 y Fp(Restart)d(Maple)f(to)g (clear)g(its)g(memory)e(of)i(all)e(de\014ned)k(v)n(ariables,)e(and)g(then)g (re-)150 1667 y(initialize)12 b(b)o(y)i(loading)e(the)j Fl(linalg)e Fp(and)h Fl(plots)g Fp(pac)o(k)n(ages.)150 1835 y Fe(Exercise)j(1)150 1918 y Fp(Imagine)g(that)j(y)o(ou)f(are)g(an)h(astronomer)e(who)h(is)h(in)o (v)o(estigating)e(the)h(orbit)h(of)e(a)150 1967 y(newly)e(disco)o(v)o(ered)g (asteroid.)24 b(Y)m(ou)16 b(w)o(an)o(t)f(to)h(determine)f(\(a\))h(what)g(is)g (the)g(closest)150 2017 y(the)g(asteroid)h(will)d(come)h(to)h(the)g(sun)g (and)g(\(b\))g(what)g(is)g(the)g(furthest)h(a)o(w)o(a)o(y)e(from)150 2067 y(the)g(sun)g(the)g(asteroid)g(will)e(get.)20 b(T)m(o)13 b(solv)o(e)h(y)o(our)h(problem,)d(y)o(ou)i(will)f(mak)o(e)g(use)j(of)150 2117 y(the)e(follo)o(wing)e(facts.)212 2170 y(a\))f(Asteroids)h(ha)o(v)o(e)e (orbits)h(whic)o(h)g(are)g(appro)o(ximately)d(elliptical,)i(with)g(the)i(sun) 150 2219 y(as)i(one)g(fo)q(cus.)212 2272 y(b\))j(In)g(p)q(olar)f(co)q (ordinates)h(an)g(ellipse)g(with)f(one)h(fo)q(cus)g(at)g(the)g(origin)e(can)i (b)q(e)150 2322 y(describ)q(ed)f(b)o(y)d(a)h(form)o(ula)d(of)j(the)g(form)637 2429 y Fj(r)e Fp(=)855 2412 y Fn(c)p 717 2419 292 2 v 717 2443 a Fi(1+)p Fn(a)6 b(sin)n Fi(\()p Fn(\022)q Fi(\)+)p Fn(b)g(cos)p Fi(\()p Fn(\022)q Fi(\))p eop bop 450 275 a Fp(62)967 b Fm(4.5)13 b(Maple)h(On)g(Line)512 391 y Fp(where)h Fl(a)p Fp(,)f Fl(b)p Fp(,)f(and)h Fl(c)g Fp(are)g(constan)o
(ts.)512 441 y(Y)m(ou)d(ha)o(v)o(e)h(also)f(collected)i(the)f(data)g(b)q(elo) o(w;)f(where)i Fj(r)g Fp(is)e(the)i(distance)f(from)e(the)450 491 y(sun)j(in)f(millions)d(of)j(miles,)f(and)i Fj(\022)h Fp(is)e(the)h (angle)f(b)q(et)o(w)o(een)i(the)f(v)o(ector)h(from)c(the)k(sun)450 541 y(to)i(the)i(asteroid)f(and)f(a)g(\014xed)h(axis)g(through)f(the)h(sun.) 27 b(The)17 b(data)f(is,)h(of)f(course,)450 591 y(sub)r(ject)f(to)f(exp)q (erimen)o(tal)f(error.)534 646 y Ff(\024)576 679 y Fj(\022)91 b Fp(0)119 b Fj(:)p Fp(6)103 b(1)p Fj(:)p Fp(8)g(1)p Fj(:)p Fp(4)g(2)p Fj(:)p Fp(1)g(3)p Fj(:)p Fp(2)g(5)p Fj(:)p Fp(4)577 729 y Fj(r)42 b Fp(329)p Fj(:)p Fp(27)e(313)p Fj(:)p Fp(8)g(319)p Fj(:)p Fp(49)g(310)p Fj(:)p Fp(91)g(327)p Fj(:)p Fp(88)g(374)p Fj(:)p Fp(91)g(367)p Fj(:)p Fp(49)1695 646 y Ff(\025)512 818 y Fp(Y)m(our)12 b(strategy)h(is)f(to)g(use)h(the)g(giv)o(en)e(data)h(to)g (\014nd)g(v)n(alues)g(of)g Fl(a)p Fp(,)g Fl(b)p Fp(,)g(and)g Fl(c)g Fp(whic)o(h)450 868 y(mak)o(e)17 b(the)h(form)o(ula)e(for)i(the)h (ellipse)f(to)g(agree)g(as)h(closely)f(as)g(p)q(ossible)h(with)e(the)450 918 y(giv)o(en)d(data..)19 b(This)14 b(will)f(in)o(v)o(olv)o(e)g(setting)i (up)f(a)g(system)g(of)g(linear)g(equations)h(in)f Fj(a)p Fp(,)450 968 y Fj(b)p Fp(,)i(and)g Fj(c)f Fp(and)h(solving)f(the)i(normal)d(equation.) 23 b(\(Giv)o(e)16 b(the)g(augmen)o(ted)f(matrices)450 1018 y(for)c(b)q(oth)h(the)g(original)d(system)j(and)f(the)h(normal)d(equation.\)) 17 b(Y)m(ou)11 b(will)f(then)i(graph)450 1067 y(the)i(giv)o(en)g(form)o(ula)d (and)j(measure)g(the)g(desired)h(data)f(from)e(the)i(graph.)512 1117 y(Note:)20 b(Once)15 b(y)o(ou)f(ha)o(v)o(e)h(found)f(v)n(alues)g(of)f (a,)h(b,)g(and)h(c,)f(y)o(ou)g(will)f(need)i(to)g(plot)450 1167 y(the)f(orbit)g(of)f(the)i(asteroid.)512 1217 y(This)h(can)f(b)q(e)h (done)g(with)f(the)h(follo)o(wing)d(command)g(where)j(r)g(is)f(the)h (function)450 1267 y(that)e(de\014nes)h(the)g(ellipse.)512 1332 y Fn(>)582 1336 y Fo(p3)21 b(:=)h(plot\([r,t,t=0..)o(2*Pi])o(,coor)o (ds=po)o(lar\):)512 1413 y Fk(Remark)p Fp(:)d(As)c(stated,)h(this)e(is)h(an)f (inheren)o(tly)h(nonlinear)f(problem)f(whic)o(h)i(w)o(e)450 1462 y(solv)o(e)k(using)g(linear)g(equations.)33 b(There)21 b(are)e(more)f(accurate)j(tec)o(hniques)f(based)450 1512 y(on)g(m)o(ultiv)n (ariabl)o(e)e(calculus.)37 b(These)21 b(tec)o(hniques)h(are)e(also)g (considerably)g(more)450 1562 y(complecated)14 b(than)f(our)h(solution.)450 1678 y Fe(Exercise)j(2)450 1755 y Fp(Maple)11 b(has)g(a)g Fk(leastsqrs)f Fp(command)e(from)i(the)i Fl(linalg)f Fp(pac)o(k)n(age)f(that)i(is)f(easier)h (\(and)450 1805 y(b)q(etter\))17 b(for)f(solving)e(least)i(squares)h (problems)d(than)i(simply)d(solving)i(the)h(normal)450 1854 y(equations.)26 b(Use)17 b(this)g(command)d(to)i(solv)o(e)g(the)i(o)o(v)o (er-determined)e(linear)g(system)450 1904 y(dev)o(elop)q(ed)e(in)f(Exercise)i (1.)j(An)c(appropriate)f(syn)o(tax)h(is)f(as)h(b)q(elo)o(w)f(where)h(A)g(is)f (the)450 1954 y(co)q(e\016cien)o(t)18 b(matrix)d(for)h(the)i(system)f(and)f Fk(F)h Fp(is)g(the)h(v)o(ector)f(of)g(constan)o(ts)h(on)e(the)450 2004 y(righ)o(t)d(side)i(of)e(the)h(equations.)512 2069 y Fn(>)582 2073 y Fo(leastsqrs\(A,F\);)p eop bop 150 703 a Fs(Chapter)31 b(5)150 910 y Fr(Determinan)m(ts)150 1126 y Fp(Determinan)o(ts)10 b(are)h(extremely)f(useful)h(in)f(man)o(y)f(con) o(texts.)18 b(Y)m(ou)10 b(will,)f(for)i(example,)150 1176 y(use)21 b(them)f(constan)o(tly)g(when)h(y)o(ou)g(study)g(eigen)o(v)n(alues)f(and)g (eigen)o(v)o(ectors)i(later)150 1226 y(in)e(the)h(text.)38 b(In)21 b(addition,)f(y)o(ou)g(will)f(see)j(them)e(used)h(to)f(write)h(form)o (ulas)d(for)150 1276 y(the)g(solutions)f(to)h(man)o(y)d(applied)i(problems.) 28 b(In)18 b(particular,)f(determinan)o(ts)h(are)150 1325 y(used)j(extensiv)o (ely)f(in)g(the)g(study)h(of)e(di\013eren)o(tial)h(equations)g(and)f(in)h (the)g(study)150 1375 y(of)c(adv)n(anced)g(calculus.)25 b(Determinan)o(ts)15 b(are)i(also)f(used)h(extensiv)o(ely)f(in)g(studying)150 1425 y(the)h(mathematical)d(foundations)i(of)g(linear)g(algebra.)26 b(Computers,)17 b(ho)o(w)o(ev)o(er,)g(do)150 1475 y(not)g(generally)g(use)i
(determinan)o(ts)e(for)g(computations.)27 b(Muc)o(h)18 b(faster)g(and)f(more) 150 1525 y(e\016cien)o(t)e(n)o(umerical)f(tec)o(hniques)i(ha)o(v)o(e)f(b)q (een)h(found.)21 b(Th)o(us,)15 b(w)o(e)g(will)e(not)i(pro)o(vide)150 1574 y(an)o(y)e(computer)h(exercises)i(for)e(this)g(c)o(hapter.)212 1624 y(The)i(reader)h(should)f(b)q(e)g(a)o(w)o(are,)f(ho)o(w)o(ev)o(er,)h (that)g(Maple)g(will)e(compute)h(deter-)150 1674 y(minan)o(ts.)h(The)e (appropriate)f(command)e(is)i(\\det\(A\);".)18 b(Inciden)o(tally)m(,)12 b(Maple)h(uses)150 1724 y(the)j(metho)q(ds)e(of)g(the)i(next)f(section)h(to)f (compute)f(determinan)o(ts)h(rather)g(than)g(the)150 1774 y(metho)q(ds)e (already)h(describ)q(ed.)804 2539 y(63)p eop bop 450 275 a Fp(64)967 b Fm(5.1)13 b(Maple)h(On)g(Line)p eop bop 150 703 a Fs(Chapter)31 b(6)150 912 y Fr(Eigen)m(v)m(ectors)150 1153 y Fq(6.1)66 b(On)22 b(Line)150 1245 y Fp(Restart)17 b(Maple)g(to)g (clear)g(its)g(memory)d(of)i(all)g(v)n(ariables,)g(and)h(re-initialize)f(it)g (b)o(y)150 1295 y(loading)c(the)j Fl(linalg)e Fp(pac)o(k)n(age.)212 1345 y(In)18 b(the)g Fk(On)i(Line)c Fp(section)i(for)f(Section)h(5.1)f(w)o(e) h(commen)o(ted)e(that)h(virtually)150 1395 y(an)o(ything)c(y)o(ou)h(migh)o(t) e(use)j(determinan)o(ts)e(for,)h(a)g(computer)f(w)o(ould)h(do)f(otherwise.) 150 1445 y(This)18 b(includes)h(\014nding)f(eigen)o(v)n(alues.)32 b(Algorithms)16 b(for)i(computing)f(eigen)o(v)n(alues)150 1495 y(are)d(v)o(ery)g(sophisticated,)g(and)f(will)f(not)i(b)q(e)g(describ)q(ed)i (in)d(this)h(text.)k(Ho)o(w)o(ev)o(er,)c(w)o(e)150 1545 y(will)f(p)q(oin)o(t) g(out)h(that)g(the)h(tec)o(hniques)h(do)d(not)h(in)o(v)o(olv)o(e)f(\014nding) h(the)h(c)o(haracteristic)150 1595 y(p)q(olynomial)i(and)k(determining)e(its) i(ro)q(ots.)38 b(In)21 b(fact,)h(the)f(n)o(umeric)f(recip)q(es)i(for)150 1644 y(\014nding)17 b(eigen)o(v)n(alues)h(are)g(so)g(go)q(o)q(d)f(that)h (they)g(are)g(often)g(used)h(to)f(\014nd)f(ro)q(ots)i(of)150 1694 y(p)q(olynomials!.)c(The)f(exercises)i(in)d(this)h(section)h(explore)f (this)g(idea.)150 1814 y Fe(Exercise)j(1)150 1906 y Fp(If)i(A)g(is)g(the)h (matrix)511 1848 y Ff(\024)568 1881 y Fp(0)73 b(1)554 1931 y Fg(\000)p Fj(b)41 b Fg(\000)p Fj(a)720 1848 y Ff(\025)742 1906 y Fp(,)20 b(sho)o(w)f(\(b)o(y)g(hand\))h(that)f(the)h(c)o(haracteristic) 150 1984 y(p)q(olynomial)9 b(is)k Fj(p)p Fp(\()p Fj(\025)p Fp(\))f(=)g Fj(\025)561 1969 y Fi(2)587 1984 y Fp(+)7 b Fj(a)g(\025)g Fp(+)h Fj(b)p Fp(.)17 b(Use)d(this)f(to)f(construct)j(a)d(matrix)f(A1)i(whic) o(h)150 2033 y(has)h Fj(p)245 2039 y Fi(1)264 2033 y Fp(\()p Fj(\025)p Fp(\))e(=)g Fj(\025)400 2018 y Fi(2)429 2033 y Fp(+)d(7)e Fj(\025)j Fp(+)f(1)14 b(as)g(its)g(c)o(haracteristic)i(p)q(olynomial.)f(Chec) o(k)g(that)f(y)o(our)150 2083 y(matrix)g(has)j(the)g(correct)h(c)o (haracteristic)g(p)q(olynomial)12 b(using)17 b(Maple's)f Fk(c)o(harp)q(oly) 150 2133 y Fp(command.)212 2184 y(Note:)30 b(The)20 b Fk(c)o(harp)q(oly)d Fp(command)g(requires)k(that)e(y)o(ou)g(name)f(the)i(v)n(ariable)150 2233 y(to)c(b)q(e)g(used)h(in)f(the)g(c)o(haracteristic)h(p)q(olynomial.)k (Th)o(us,)16 b(an)g(appropriate)g(syn)o(tax)150 2283 y(migh)o(t)c(b)q(e)212 2350 y Fn(>)282 2354 y Fo(p_1:=charpoly\(A,)o(lambd)o(a\);)212 2434 y Fp(Also,)f(Maple)f(uses)h Fj(p)p Fp(\()p Fj(\025)p Fp(\))h(=)g Fj(det)p Fp(\()p Fj(\025)7 b(I)e Fg(\000)r Fj(A)p Fp(\))p Fj(p)p Fp(\()p Fj(\025)p Fp(\))13 b(=)f Fj(det)p Fp(\()p Fj(A)r Fg(\000)r Fj(\025)7 b(I)s Fp(\))12 b(as)e(the)h(de\014nition)804 2539 y(65)p eop bop 450 275 a Fp(66)967 b Fm(6.1)13 b(Maple)h(On)g(Line)450 391 y Fp(of)i(the)i(c)o(haracteristic)h(p)q(olynomial)o(.)25 b(The)17 b(relation)g(b)q(et)o(w)o(een)h(the)g(t)o(w)o(o)f(is)g(that)g(if)450 441 y(A)e(is)f(\(n)h(x)f(n\),)h(then)g(the)g(c)o(haracteristic)h(p)q (olynomial)c(w)o(e)i(compute)h(in)f(the)h(text)g(is)450 491 y(\()p Fg(\000)p Fp(1\))535 476 y Fn(n)512 545 y Fp(Use)e(Maple's)f Fk(eigen)o(v)m(als)e Fp(command)f(to)j(compute)g(the)h(eigen)o(v)n(alues)f (of)f(A1)h(and)450 594 y(hence)j(the)g(ro)q(ots)f(of)f Fj(p)810 600 y Fi(1)829 594 y Fp(\()p Fj(\025)p Fp(\))512 648 y(Note:)19
b(An)14 b(appropriate)g(syn)o(taxt)g(for)f(the)i(solv)o(e)e(command)f(is)512 724 y Fn(>)582 728 y Fo(solve\(p1)20 b(=)i(0,lambda\);)450 865 y Fe(Exercise)17 b(2)450 988 y Fp(Compute)8 b(the)i(c)o(haracteristic)h (p)q(olynomial)6 b(for)j(the)g(matrix)f(A)h(=)1491 904 y Ff(2)1491 979 y(4)1554 937 y Fp(0)71 b(1)h(0)1554 987 y(0)f(0)h(1)1539 1037 y Fg(\000)p Fj(c)42 b Fg(\000)p Fj(b)g Fg(\000)p Fj(a)1798 904 y Ff(3)1798 979 y(5)1825 988 y Fp(.)450 1090 y(Use)16 b(this)e(result)i (to)e(obtain)g(the)i(ro)q(ots)f(of)f(the)h(p)q(olynomial)26 b Fj(p)1460 1096 y Fi(1)1478 1090 y Fp(\()p Fj(\025)p Fp(\))14 b(=)f Fj(\025)1617 1075 y Fi(3)1645 1090 y Fp(+)d(8)d Fj(\025)1739 1075 y Fi(2)1768 1090 y Fp(+)450 1140 y(17)g Fj(\025)i Fp(+)g(10.)18 b(T)m(est)c(the)h(ro)q(ots)f(b)o(y)g(substitution)g(bac)o(k)g(in)o(to)f Fj(p)p Fp(\()p Fj(\025)p Fp(\).)450 1276 y Fe(Exercise)k(3)450 1360 y Fp(Let)g(A1)f(b)q(e)h(the)g(matrix)e(obtained)h(in)g(Exercise)i(3.)26 b(Obtain)16 b(the)h(eigen)o(v)o(ectors)h(of)450 1409 y(A1,)c(normalizing)e (them)i(so)g(the)i(\014rst)f(elemen)o(t)f(in)g(eac)o(h)h(is)g(1.)20 b(What)14 b(do)g(y)o(ou)g(then)450 1459 y(notice)i(ab)q(out)g(these)h(eigen)o (v)o(ectors?)25 b(Use)17 b(the)f(pattern)h(y)o(ou)e(articulate)h(to)g(giv)o (e)f(a)450 1509 y(general)e(prescription)h(of)e(the)i(eigen)o(v)o(ectors)g (of)e(an)h(\(n)g(x)f(n\))h(matrix)e(of)i(the)g(form)e(A.)450 1559 y(Pro)o(v)o(e)j(y)o(our)g(answ)o(er.)512 1612 y(The)d(eigen)o(v)n (actors)f(for)f(A1)h(ma)o(y)e(b)q(e)i(computed)g(using)f(the)i(follo)o(wing)c (command.)512 1689 y Fn(>)582 1693 y Fo(q)21 b(:=)h(eigenvects\(A2\);)512 1785 y Fp(Note)e(that)f(there)h(are)g(three)g(lists)f(in)g(q,)h(and)f(in)f (eac)o(h)i(list)f(there)h(are)g(three)450 1834 y(mem)o(b)q(ers.)34 b(The)20 b(\014rst)g(mem)o(b)q(er)e(of)h(eac)o(h)h(list)f(is)g(the)h(eigen)o (v)n(alue.)35 b(The)20 b(second)450 1884 y(mem)o(b)q(er)13 b(of)g(eac)o(h)i(list)f(is)g(the)h(algebraic)f(m)o(ultipli)o(cit)o(y)m(,)d (the)j(n)o(um)o(b)q(er)g(of)g(times)f(that)450 1934 y(eigen)o(v)n(alue)k(is)g (a)g(ro)q(ot)g(of)g(the)h(c)o(haracteristic)h(equation.)28 b(The)18 b(third)f(mem)o(b)q(er)f(of)450 1984 y(eac)o(h)k(list)g(is)f(a)h (set)g(of)g(eigen)o(v)o(ectors.)37 b(Here,)22 b(eac)o(h)e(suc)o(h)g(set)h (con)o(tains)f(a)f(single)450 2034 y(eigen)o(v)o(ector.)g(Hence,)c(these)g (eigen)o(v)o(ectors)g(can)f(b)q(e)h(referenced)h(as)e(follo)o(ws.)512 2110 y Fn(>)582 2114 y Fo(v1)21 b(:=)h(q[1][3][1];)d(v2)i(:=)h(q[2][3][1];)d (v3)j(:=)f(q[3][3][1];)450 2250 y Fe(Exercise)c(4)450 2334 y Fp(Find)11 b(a)g(4)h(x)f(4)g(matrix)f(A)h(whose)i(c)o(haracteristic)f(p)q (olynomial)d(is)i Fj(p)p Fp(\()p Fj(\025)p Fp(\))h(=)g Fj(\025)1633 2319 y Fi(4)1656 2334 y Fp(+)t(3)7 b Fj(\025)1744 2319 y Fi(2)1768 2334 y Fg(\000)450 2384 y Fp(5)g Fj(\025)j Fp(+)h(7.)22 b(Obtain)15 b(the)h(ro)q(ots)f(of)g(this)h(p)q(olynomial)c(b)o(y)j(\014nding)f(the)i (eigen)o(v)n(alues)g(of)450 2434 y(the)e(matrix)f(A.)g(Chec)o(k)i(y)o(our)e (result)i(with)e(Maple's)h Fk(solv)o(e)f Fp(command.)p eop bop 150 275 a Fm(6.2)13 b(Maple)h(On)g(Line)967 b(67)212 391 y Fp(The)16 b(matrix)d(whose)j(c)o(haracteristic)g(p)q(olynomial)c(is)j Fj(p)p Fp(\()p Fj(\025)p Fp(\))g(is)g(kno)o(wn)g(in)g(math-)150 441 y(ematics)f(as)g(the)i(companion)c(matrix.)18 b(Maple)c(has)h(the)g (built-in)e(command)f Fk(com-)150 491 y(panion)h Fp(for)h(\014nding)h(the)g (companion)e(matrix)g(for)i(a)g(p)q(olynomial)o(.)j(Maple)d(gener-)150 541 y(ates)i(a)f(companion)e(matrix)g(that)i(is)g(the)h(transp)q(ose)h(of)d (what)h(y)o(ou)g(migh)o(t)e(\014nd)i(in)150 591 y(some)d(di\013eren)o(tial)h (equations)g(texts.)212 658 y Fn(>)282 662 y Fo(p)21 b(:=)h(x^4)f(+)h(3*x^2)e (-)i(5*x)f(+)h(7;)212 729 y Fn(>)282 733 y Fo(A)f(:=)h(transpose\(compan)o (ion\(p)o(,x\)\);)212 812 y Fp(Find)14 b(the)g(eigen)o(v)n(alues)g(of)f(A.) 212 879 y Fn(>)282 883 y Fo(eigenvals\(A\);)212 962 y Fp(Maple)20 b(has)g(expressed)j(the)d(ro)q(ots)h(of)e(the)i(c)o(haracteristic)h(p)q (olynomial)16 b(with)150 1011 y(its)h("Ro)q(otOf")g(notation.)28 b(This)17 b(is)g(a)g(shorthand)h(for)f(what)h(could)f(b)q(e)h(large)f(and)150 1061 y(complex)h(expressions)i(for)f(the)h(exact)f(v)n(alue)g(of)f(the)i(ro)q
(ots.)34 b(There)20 b(are)f(sev)o(eral)150 1111 y(options)g(a)o(v)n(ailable)e (at)h(this)i(p)q(oin)o(t.)33 b(First,)20 b(apply)e(the)i Fk(allv)m(alues)d Fp(command)f(to)150 1161 y(the)k(Ro)q(otOf)f(structure.)37 b(This)20 b(will)e(return)i(the)h(eigen)o(v)n(alues)e(as)g(exact)i(v)n (alues,)150 1211 y(con)o(taining)13 b(complicated)f(expressions)k(in)o(v)o (olving)11 b(radicals.)212 1278 y Fn(>)282 1282 y Fo(q)21 b(:=)h (eigenvals\(A\);)212 1349 y Fn(>)282 1353 y Fo(q1)f(:=)h(allvalues\(q\);)212 1432 y Fp(These)f(expressions)g(are)g(to)q(o)e(un)o(wieldy)g(to)h(w)o(ork)f (with.)35 b(Con)o(v)o(ert)20 b(them)f(to)150 1481 y(\015oating)13 b(p)q(oin)o(t)g(n)o(um)o(b)q(ers)h(with)f(the)i Fk(ev)m(alf)e Fp(command.)212 1548 y Fn(>)282 1552 y Fo(evalf\(q1\);)212 1631 y Fp(Another)j(alternativ)o(e)f(is)g(to)g(include)g(at)g(least)h(one)f (\015oating)f(p)q(oin)o(t)h(n)o(um)o(b)q(er)f(in)150 1681 y(the)c(matrix)d (A.)i(When)h Fk(eigen)o(v)m(als)d Fp(sees)k(the)f(\015oat,)g(it)e(will)g (compute)h(the)h(eigen)o(v)n(alues)150 1731 y(n)o(umerically)m(.)212 1798 y Fn(>)282 1802 y Fo(A1)21 b(:=)h(map\(convert,A,f)o(loat\))o(;)212 1869 y Fn(>)282 1873 y Fo(eigenvals\(A1\);)212 1952 y Fp(Finally)m(,)17 b(solv)o(e)g(the)i(equation)e Fj(p)p Fp(\()p Fj(\025)p Fp(\))i(=)f(0)g(n)o (umerically)e(b)o(y)i(using)f(the)i Fk(fsolv)o(e)150 2002 y Fp(command.)c(F)m(or)c(p)q(olynomial)e(equations,)j(this)g(command)d(accepts) 14 b(the)e(parameter)150 2052 y("complex")c(to)i(indicate)g(that)g(all)e(ro)q (ots,)j(b)q(oth)f(real)g(and)f(complex,)g(are)i(to)e(b)q(e)i(found.)212 2119 y Fn(>)282 2123 y Fo(fsolve\(p,x,compl)o(ex\);)150 2292 y Fq(6.2)66 b(On)22 b(Line)150 2384 y Fp(Restart)f(Maple)g(to)g(clear)g(its)g (memory)d(of)i(all)g(v)n(ariables,)h(then)g(reinitialize)f(b)o(y)150 2434 y(loading)12 b(the)j Fl(linalg)e Fp(pac)o(k)n(age.)p eop bop 450 275 a Fp(68)967 b Fm(6.2)13 b(Maple)h(On)g(Line)512 391 y Fp(Create)20 b(y)o(our)f(o)o(wn)g(eigen)o(v)n(alue)f(problem)g(b)o(y)h (constructing)h(a)e(3)h(x)g(3)g(matrix)450 441 y(A)f(with)f(prescrib)q(ed)i (eigen)o(v)n(alues)f(and)f(eigen)o(v)o(ectors.)30 b(The)18 b(eigen)o(v)n(alues)f(are)h(the)450 491 y(diagonal)d(elemen)o(ts)i(in)g(a)f (diagonal)f(matrix)g(D,)i(while)f(the)i(eigen)o(v)o(ectors)g(are)f(the)450 541 y(columns)c(of)g(a)h(3)f(x)h(3)f(matrix)f(P)m(.)512 594 y(A)20 b(simple)e(w)o(a)o(y)g(to)h(construct)i(the)f(eigen)o(v)o(ector)h (matrix)c(P)j(is)f(as)g(a)h(random)450 644 y(matrix.)c(Hence,)f(de\014ne)f (f,)f(a)g(function)g(whic)o(h)g(generates)i(random)d(in)o(tegers)i(in)f(the) 450 694 y(in)o(terv)n(al)g([-10,10].)512 770 y Fn(>)582 774 y Fo(f)21 b(:=)h(rand\(-10..10\):)512 864 y Fp(Let)15 b(P)f(b)q(e)g(a)g (random)e(3)i(x)f(3)h(matrix)e(with)i(en)o(tries)h(determined)f(b)o(y)f(f.) 512 940 y Fn(>)582 944 y Fo(P)21 b(:=)h(randmatrix\(3,3,e)o(ntrie)o(s=f\);) 512 1035 y Fp(Let)d(the)g(eigen)o(v)n(alues)f(b)q(e)h(2,)f(2,)h(and)f(3,)g (in)g(that)g(order.)32 b(Create)19 b(a)f(diagonal)450 1085 y(matrix)c(with)h(these)j(elemen)o(ts)d(on)h(the)h(diagonal,)d(but)i(assign)f (the)i(matrix)d(to)h(the)450 1135 y(name)h(d,)i(not)g(D.)f(The)h(letter)g ("D")f(in)g(Maple)g(is)h(reserv)o(ed)i(for)d(one)h(form)d(of)i(the)450 1184 y(di\013eren)o(tiation)d(op)q(erator)g(and)g(Maple)f(will)g(not)h(let)g (y)o(ou)f(assign)h(to)f(it.)512 1260 y Fn(>)582 1264 y Fo(d)21 b(:=)h(diag\(2,2,3\);)512 1355 y Fp(The)17 b(matrix)d Fj(A)h Fp(=)g Fj(P)e(D)8 b(P)945 1340 y Fi(\()p Fh(\000)p Fi(1\))1031 1355 y Fp(will)15 b(ha)o(v)o(e)h(eigen)o(v)n(alues)f(2,)h(2,)g(and)g(3)g(-)f (in)h(that)450 1405 y(order)i(-)f(and)g(will)f(ha)o(v)o(e)h(the)g(columns)f (of)h(P)g(as)h(eigen)o(v)o(ectors,)g(in)f(corresp)q(onding)450 1455 y(order.)512 1530 y Fn(>)582 1534 y Fo(A)k(:=)h(evalm\(P)e(&*)h(d)h(&*)f (inverse\(P\)\);)450 1707 y Fe(Exercise)c(1)450 1790 y Fp(By)h(computing)f(A) h Fk(X)g Fp(for)f(eac)o(h)i(column)d Fk(X)i Fp(of)g(P)m(,)f(v)o(erify)h(that) g(eac)o(h)g(column)f(of)450 1840 y(P)f(is)g(an)f(eigen)o(v)o(ector)i(of)e(A.) h(Columns)e(of)h(P)h(can)g(b)q(e)g(referenced)j(b)o(y)c(Maple's)h Fk(col)450 1890 y Fp(command.)f(Clearly)m(,)c(A)i Fk(X)f Fp(=)h Fj(\025)g Fk(X)g Fp(m)o(ust)e(hold)h(for)h(eac)o(h)g(eigenpair)f(of)g(eigen)o
(v)o(ector)450 1940 y Fk(X)i Fp(and)g(eigen)o(v)n(alue)f Fj(\025)p Fp(.)450 2112 y Fe(Exercise)k(2)450 2195 y Fp(V)m(erify)f(that)h(the)g (diagonal)e(elemen)o(ts)i(of)f(D)h(are)g(the)g(eigen)o(v)n(alues)g(of)f(A)h (b)o(y)f(using)450 2245 y(Maple's)j Fk(eigen)o(v)m(als)e Fp(command)g(to)i (determine)g(the)h(eigen)o(v)n(alues)f(of)g(A)h(directly)450 2295 y(from)15 b(A)i(itself.)26 b(There)18 b(is,)f(ho)o(w)o(ev)o(er,)g(no)g (canonical)f(ordering)h(for)f(the)i(results)g(of)450 2345 y(this)c(command,)d (so)j(Maple)f(need)i(not)f(order)h(the)f(eigen)o(v)n(alues)g(as)g(2,)f(2,)g (3.)512 2420 y Fn(>)582 2424 y Fo(eigenvals\(A\);)p eop bop 150 275 a Fm(6.2)13 b(Maple)h(On)g(Line)967 b(69)150 391 y Fe(Exercise)17 b(3)150 469 y Fp(By)k(applying)e(Maple's)h Fk(eigen)o(v)o(ects)e Fp(command)g(to)j(A,)f(again)f(v)o(erify)i(that)f(the) 150 519 y(columns)12 b(of)h(P)h(are)g(the)g(eigen)o(v)o(ectors.)19 b(The)14 b Fk(eigen)o(v)o(ects)c Fp(command)h(returns)k(lists)150 569 y(with)f(three)h(mem)o(b)q(ers)d(in)i(eac)o(h)g(list.)k(These)d(three)g (mem)o(b)q(ers)e(are)h(\014rst,)g(the)h(eigen-)150 619 y(v)n(alue;)g(second,) h(the)g(algebraic)e(m)o(ultiplicit)o(y)e(of)j(the)h(eigen)o(v)n(alue)e(-)h (the)h(n)o(um)o(b)q(er)f(of)150 668 y(times)d(the)i(eigen)o(v)n(alue)f(w)o (as)g(a)h(ro)q(ot)f(of)g(the)h(c)o(haracteristic)g(equation;)f(and)g(third,)g (a)150 718 y(set)i(of)e(eigen)o(v)o(ectors)i(asso)q(ciated)g(with)e(the)i (eigen)o(v)n(alue)e(in)h(the)g(list.)212 769 y(Extract)k(the)g(eigen)o(v)n (alues)e(and)h(eigen)o(v)o(ectors)h(b)o(y)f(adroit)f(use)i(of)e(the)i (selector)150 818 y(brac)o(k)o(et)g(notation)f([].)27 b(\(See)19 b(Exercise)g(3)e(in)g(the)h(On)g(Line)f(exercises)i(for)f(Section)150 868 y(6.1.\))f(Again,)11 b(there)j(is)e(no)g(canonical)f(ordering)i(for)f (the)h(lists)f(pro)q(duced,)h(or)f(for)g(the)150 918 y(eigen)o(v)o(ectors)18 b(asso)q(ciated)g(with)f(an)g(eigen)o(v)n(alue)g(of)f(m)o(ultiplicit)o(y)e (greater)19 b(than)e(1.)150 968 y(Executing)e(the)g Fk(eigen)o(v)o(ects)d Fp(command)g(on)i(di\013eren)o(t)h(o)q(ccasions)g(can)g(result)g(in)f(a)150 1018 y(di\013eren)o(t)h(ordering)f(eac)o(h)g(time.)212 1068 y(The)e(eigen)o(v)o(ectors)g(computed)f(b)o(y)g(the)h Fk(eigen)o(v)o(ects)c Fp(command)h(ma)o(y)g(not)i("lo)q(ok)150 1118 y(lik)o(e")h(the)i(columns)e (of)g(P)m(.)g(The)i(columns)e(of)g(P)h(ma)o(y)e(b)q(e)j(constan)o(t)g(m)o (ultiples)d(of)h(the)150 1168 y(corresp)q(onding)g(v)o(ectors)h(,)e(or,)h(in) e(the)i(case)h(of)e(m)o(ultiple)e(eigen)o(v)o(ectors,)j(the)g(columns)150 1218 y(of)f(P)h(could)g(simply)d(b)q(e)k(a)e(di\013eren)o(t)i(basis)f(for)f (the)h(eigenspace)h(asso)q(ciated)g(with)e(the)150 1267 y(rep)q(eated)16 b(eigen)o(v)n(alue.)21 b(Compare)13 b(the)i(third)g(column)e(of)h(P)h(with)g (the)g(eigen)o(v)o(ector)150 1317 y(Maple)k(found)g(for)h(the)g(eigen)o(v)n (alue)f(3,)h(determining)e(an)o(y)i(m)o(ultipli)o(cativ)o(e)d(factor)150 1367 y(needed)e(to)f(mak)o(e)e(the)j(eigen)o(v)o(ectors)g(matc)o(h)e(exactly) m(.)212 1417 y(T)m(o)h(sho)o(w)h(t)o(w)o(o)f(bases)h Fg(f)f Fj(P)628 1423 y Fi(1)647 1417 y Fj(;)f(P)699 1423 y Fi(2)717 1417 y Fg(g)h Fp(and)h Fg(f)f Fj(v)889 1423 y Fi(1)908 1417 y Fj(;)f(v)953 1423 y Fi(2)972 1417 y Fg(g)h Fp(are)h(equiv)n(alen)o(t,)f(y)o (ou)g(need)h(to)150 1467 y(sho)o(w)9 b(that)h(linear)f(com)o(binations)e(of)i (one)h(set)g(of)f(basis)g(v)o(ectors)i(yield)e(the)h(other)g(basis)150 1517 y(v)o(ectors.)20 b(Use)c(the)e Fk(rref)f Fp(command)f(on)i(a)g(matrix)e (con)o(taining)i(as)g(its)g(columns)f(the)150 1567 y(\014rst)h(t)o(w)o(o)e (columns)g(of)g(P)h(and)g(the)h(eigen)o(v)o(ectors)g(Maple)f(found)f(for)h (the)h(eigen)o(v)n(alue)150 1617 y(2.)k(Ho)o(w)13 b(will)g(this)h(sho)o(w)g (the)g(equiv)n(alence)g(of)f(the)i(bases?)150 1736 y Fe(Exercise)i(4)150 1814 y Fp(F)m(or)11 b(the)h Fl(n)g Fp(x)f Fl(n)h Fp(matrix)e(A,)h(the)h(c)o (haracteristic)h(p)q(olynomial)8 b Fj(p)p Fp(\()p Fj(\025)p Fp(\))k(has)f(b)q(een)i(de\014ned)150 1864 y(in)i(this)h(text)g(as)f Fj(p)p Fp(\()p Fj(\025)p Fp(\))g(=)g Fj(det)o Fp(\()p Fj(A)c Fg(\000)g Fj(\025)c(I)s Fp(\).)23 b(Some)15 b(texts)h(use)h Fj(det)o Fp(\()p Fj(\025)7 b(I)15 b Fg(\000)10 b Fj(A)p Fp(\),)16 b(thereb)o(y)150 1914 y(making)g(the)j(t)o(w)o(o)g(de\014nitions)f(di\013er)h
(b)o(y)g(a)f(factor)h(of)f(\()p Fg(\000)p Fp(1\))1141 1899 y Fn(n)1164 1914 y Fp(.)32 b(Maple's)18 b(built-in)150 1963 y Fk(c)o(harp)q(oly)c Fp(command)f(for)i(generating)h(the)g(c)o (haracteristic)h(p)q(olynomial)12 b(uses)17 b(the)150 2013 y(latter)e(con)o(v)o(en)o(tion.)22 b(The)16 b(adv)n(an)o(tage)e(of)h(Maple's) g(de\014nition)g(is)g(that)g(for)g(an)g(\(n)g(x)150 2063 y(n\))f(matrix,)e (the)i(highest)g(order)h(term)212 2114 y(In)h(Maple,)g(compare)f(these)j(t)o (w)o(o)d(metho)q(ds)h(for)f(obtaining)g(the)i(c)o(haracteristic)150 2163 y(p)q(olynomial.)c(The)e(c)o(harp)q(oly)f(command)e(tak)o(es)i(as)h (argumen)o(ts)e(the)i(matrix)e(A)h(and)g(a)150 2213 y(v)n(ariable)e(to)i(b)q (e)g(used)g(in)f(the)h(output)g(p)q(olynomial)o(.)k(Maple)9 b(computes)g(determinan)o(ts)150 2263 y(via)k(the)h Fk(det)e Fp(command,)f(and)i(also)g(allo)o(ws)f(the)i(syn)o(tax)g Fj(A)8 b Fg(\000)h Fj(\025)14 b Fp(as)f(a)h(short)g(form)e(of)150 2313 y Fj(A)7 b Fg(\000)g Fj(\025)g(I)s Fp(.)18 b(Finally)m(,)10 b(t)o(yping)i(out)h(the)g("name")e(of)h(the)h(Greek)h(letter)f Fj(\025)g Fp(causes)h(Maple)150 2363 y(to)g(prin)o(t)f(that)h(letter)h(as)f (a)g(Greek)g(letter.)212 2430 y Fn(>)282 2434 y Fo(charpoly\(A,lambd)o(a\);)p eop bop 450 275 a Fp(70)967 b Fm(6.3)13 b(Maple)h(On)g(Line)512 387 y Fn(>)582 391 y Fo(det\(A-lambda\);)512 479 y Fp(Note)h(that)f(the)g(t)o (w)o(o)f(p)q(olynomials)e(are)k(just)f(negativ)o(es)g(of)f(eac)o(h)h(other.) 512 532 y(There)i(is)f(only)f(one)h(degree)i(three)f(p)q(olynomial)11 b(with)k(ro)q(ots)g(2,)f(2,)h(and)g(3,)f(that)450 582 y(has)f Fj(\025)547 567 y Fi(3)566 582 y Fp(as)g(its)f(highest)h(degree)i(term.)i (What)12 b(is)h(this)f(p)q(olynomial?)j(\(Hin)o(t:)i(W)m(rite)12 b(it)450 632 y(as)i(a)g(pro)q(duct)g(of)g(linear)f(factors)h(and)g(then)h (expand.\))512 684 y(Note)c(that)e(this)h(is)g(the)g(c)o(haracteristic)i(p)q (olynomial)6 b(of)j($A$)h(as)g(found)f(b)o(y)h(Maple.)450 837 y Fq(6.3)66 b(On)22 b(Line)450 933 y Fp(Restart)14 b(Maple)f(to)g(clear)g (its)g(memory)e(of)i(all)f(de\014ned)i(v)n(ariables,)e(and)h(re-initialize) 450 983 y(b)o(y)h(loading)e(the)i Fl(linalg)g Fp(pac)o(k)n(age.)512 1036 y(In)h(these)i(exercises)g(complex)d(n)o(um)o(b)q(ers)h(will)f(app)q (ear.)22 b(Maple)15 b(uses)h(the)g(letter)450 1086 y("I")c(for)581 1053 y Fg(p)p 615 1053 54 2 v 615 1086 a(\000)p Fp(1,)g(so)h(that)f(the)h (complex)e(n)o(um)o(b)q(er)g(z)i(=)f(2)g(+)h(3)f Fl(i)g Fp(is)g(en)o(tered)i (in)o(to)d(Maple)450 1135 y(as)18 b(z)h(=)f(2)g(+)g(3*I.)f(It)h(is)g(also)f (useful)h(to)g(remem)o(b)q(er)f(that)h(if)g(z)g(=)g(2)g(+)g(3)g Fl(i)p Fp(,)g(then)p 450 1162 22 2 v 450 1185 a Fj(z)c Fp(=)d(2)c Fg(\000)g Fp(3)g Fj(i)12 b Fp(is)h(the)g(complex)e(conjugate)i(of)f(z.)18 b(Th)o(us,)12 b(the)h(complex)f(conjugate)g(of)g(a)450 1235 y(real)h(n)o(um)o(b)q(er)f(x)g(is)h(that)f(real)h(n)o(um)o(b)q(er)f(itself,)g (since)h(the)h(imaginary)c(part)i(\(the)i(part)450 1285 y(with)g(the)g Fl(i)p Fp(\))g(is)f(zero.)512 1338 y(Maple's)18 b(command)e(for)i (conjugating)g(a)g(n)o(um)o(b)q(er)f(is)i Fk(conjugate)p Fp(.)29 b(Its)19 b(com-)450 1387 y(mands)d(for)i(extracting)g(the)g(real)g(and)f (imaginary)e(parts)j(of)f(a)g(complex)g(n)o(um)o(b)q(er)450 1437 y(are)e Fk(Re)f Fp(and)g Fk(Im)p Fp(,)g(resp)q(ectiv)o(ely)m(.)20 b(In)15 b(purely)f(n)o(umeric)g(con)o(texts)i(these)f(commands)450 1487 y(usually)g(need)j(no)e(additional)e(b)q(o)q(osts.)26 b(In)16 b(sym)o(b)q(olic)f(and)h(exact)h(con)o(texts,)h(these)450 1537 y(commands)10 b(generally)h(w)o(ork)h(only)g(if)f(an)h(additional)e Fk(ev)m(alc)i Fp(\(ev)n(aluate)g(complex\))f(is)450 1587 y(applied.)17 b(Th)o(us,)512 1660 y Fn(>)582 1664 y Fo(Re\(2)k(+)g(3*I\);)g(Im\(2)g(+)h (3*I\);)512 1753 y Fp(but)512 1826 y Fn(>)582 1830 y Fo(Re\(x)f(+)g(I*y\);)g (Im\(x)g(+)h(I*y\);)512 1919 y Fp(thereb)o(y)15 b(requiring)512 1992 y Fn(>)582 1996 y Fo(evalc\(Re\(x)20 b(+)h(I*y\)\);)g(evalc\(Im\(x)e(+)j (I*y\)\);)450 2163 y Fe(Exercise)17 b(1)450 2245 y Fp(Use)d(Maple's)f Fk(eigen)o(v)o(ects)e Fp(command)f(to)j(obtain)g(the)g(eigen)o(v)n(alues)g
(and)g(eigen)o(v)o(ec-)450 2318 y(tors)i(of)g(the)g(matrix)f(A)h(=)885 2259 y Ff(\024)928 2292 y Fp(1)41 b Fg(\000)p Fp(3)928 2342 y(1)57 b(1)1064 2259 y Ff(\025)1101 2318 y Fp(from)13 b(Example)g(2)i(of)f (section)i(5.3.)21 b(Notice)450 2390 y(that)11 b(the)h(eigen)o(v)n(alues)f (are)g(complex)f(conjugates,)h(as)h(are)f(the)h(eigen)o(v)o(ectors)g(as)f(w)o (ell.)p eop bop 150 275 a Fm(6.3)13 b(Maple)h(On)g(Line)967 b(71)150 391 y Fe(Exercise)17 b(2)150 473 y Fp(An)i Fl(n)g Fp(x)g Fl(n)g Fp(matrix)e(A)i(with)f(complex)g(en)o(tries)i(is)e(said)h(to)g(b)q(e)g (Hermitian)e(if)h(the)150 522 y(conjugate)i(of)g(the)h(transp)q(ose)g(equals) f(A.)g(Th)o(us,)i(A)e(is)g(Hermitian)f(if)p 1335 487 46 2 v 19 w Fj(A)1366 510 y Fn(t)1403 522 y Fp(=)j Fj(A)p Fp(.)150 572 y(A)17 b(momen)o(t's)e(re\015ection)j(rev)o(eals)g(that)f(the)h (conjugate)g(of)e(the)i(transp)q(ose)h(equals)150 629 y(the)e(transp)q(ose)i (of)d(the)h(conjugate,)g(that)g(is,)p 891 594 V 17 w Fj(A)922 617 y Fn(t)953 629 y Fp(=)p 1002 595 32 2 v 17 w Fj(A)1033 602 y Fn(t)1048 629 y Fp(.)26 b(Some)16 b(texts)i(denote)f(the)150 685 y(conjugate)e(transp)q(ose)h(of)e(A)g(b)o(y)h(the)g(sym)o(b)q(ol)e Fj(A)924 670 y Fh(\003)958 685 y Fp(so)h(that)h Fj(A)1131 670 y Fh(\003)1163 685 y Fp(=)p 1208 652 V 13 w Fj(A)1239 659 y Fn(t)1268 685 y Fp(=)p 1315 650 46 2 v 15 w Fj(A)1346 673 y Fn(t)1361 685 y Fp(,)f(and)h(A)150 735 y(is)f(Hermitian)e(pro)o(vided)i Fj(A)d Fp(=)h Fj(A)677 720 y Fh(\003)697 735 y Fp(.)212 787 y(Giv)o(e)18 b(an)g(example)f(of)g(a)h(3)g(x)g(3)g(Hermitian)f(matrix)f(con)o (taining)h(as)i(few)f(real)150 837 y(n)o(um)o(b)q(ers)c(as)g(p)q(ossible,)f (and)h(ha)o(ving)f(no)g(en)o(tries)i(zero.)212 889 y(T)m(o)k(v)o(erify)g (that)h(y)o(our)f(matrix)f(is)h(Hermitian,)g(y)o(ou)g(need)i(to)e(tak)o(e)h (b)q(oth)f(the)150 939 y(transp)q(ose)c(and)f(the)h(complex)e(conjugate.)18 b(Maple)c(has)g(the)h(built-in)e Fk(h)o(transp)q(ose)150 989 y Fp(command)g(for)i(this.)23 b(Apply)16 b(it)f(to)g(y)o(our)h(matrix)d(A,)j (and)f(then)h(separately)h(apply)150 1039 y(the)e Fk(transp)q(ose)d Fp(and)j Fk(conjugate)d Fp(commands.)17 b(In)d(Maple,)g(the)h(simplest)e(w)o (a)o(y)h(to)150 1089 y(do)g(this)g(is)f(to)h(map)e Fk(conjugate)g Fp(on)o(to)i(the)g(transp)q(ose)i(of)d(A.)212 1161 y Fn(>)282 1165 y Fo(htranspose\(A\);)212 1238 y Fn(>)282 1242 y Fo(map\(conjugate,tr)o (anspo)o(se\(A\))o(\);)212 1328 y Fp(One)f(remark)n(able)e(prop)q(ert)o(y)i (of)e(Hermitian)f(matrices)i(is)g(that)g(their)g(eigen)o(v)n(alues)150 1378 y(are)j(real.)j(Apply)c(Maple's)g Fk(eigen)o(v)m(als)e Fp(command)f(to)j(y)o(our)g(matrix)e(A)j(in)e(an)h(e\013ort)150 1428 y(to)h(v)o(erify)f(the)h(truth)g(of)g(this)f(claim.)j(Y)m(ou)d(will)f (most)h(lik)o(ely)f(obtain)h(large,)g(compli-)150 1477 y(cated)i(expressions) h(for)d(the)i(sym)o(b)q(olically)c(exact)k(eigen)o(v)n(alues.)k(They)14 b(migh)o(t)e(ev)o(en)150 1527 y(con)o(tain)17 b(the)h(sym)o(b)q(ol)d("I",)i (making)e(it)i(lo)q(ok)f(lik)o(e)h(the)h(eigen)o(v)n(alues)f(are)h(complex!) 150 1577 y(T)m(o)13 b(determine)g(if)g(these)i(eigen)o(v)n(alues)f(are)g (real,)f(y)o(ou)g(w)o(an)o(t)g(to)g(sho)o(w)h(that)g(for)f(eac)o(h,)150 1627 y(the)h(imaginary)d(part)k(is)e(zero.)212 1679 y(After)19 b(obtaining)e(the)i(exact)f(eigen)o(v)n(alues,)h(con)o(v)o(ert)g(them)e(to)h (\015oating)f(p)q(oin)o(t)150 1729 y(n)o(um)o(b)q(ers)12 b(b)o(y)h(using)f (the)h Fk(ev)m(alf)f Fp(command.)j(Since)e Fk(eigen)o(v)m(als)d Fp(returns)15 b(a)d(sequence)150 1779 y(of)e(eigen)o(v)n(alues,)h(y)o(ou)f (ma)o(y)f(need)j(to)f(con)o(v)o(ert)g(this)g(return)h(to)f(a)g(list)f(\(use)i ([...]\))j(b)q(efore)150 1829 y(the)i Fk(ev)m(alf)e Fp(command)e(w)o(orks.)24 b(The)17 b(con)o(v)o(ersion)f(to)f(decimals)g(of)g(an)o(y)h(radicals)f(in)150 1878 y(y)o(our)j(eigen)o(v)n(alues)g(ma)o(y)f(yield)h(expressions)i(with)e(v) o(ery)g(small)e(imaginary)g(parts.)150 1928 y(Y)m(ou)f(can)g(instruct)i (Maple)e(to)g(truncate)i(these)g(small)c(n)o(um)o(b)q(ers)i(to)g(zero)h(b)o (y)f(using)150 1978 y(the)f Fk(fnormal)e Fp(command.)212 2030 y(Instruct)h(Maple)e(to)g(extract)h(and)f(simplify)d(the)k(imaginary)c(part)k
(of)e(eac)o(h)i(eigen-)150 2080 y(v)n(alue.)17 b(These)e(imaginary)c(parts)k (should)e(reduce)j(exactly)d(to)h(zero.)19 b(Remem)o(b)q(er)12 b(to)150 2130 y(use)j(b)q(oth)f Fk(ev)m(alc)g Fp(and)f Fk(Im)p Fp(,)h(as)g(w)o(ell)f(as)h Fk(simplif)o(y)p Fc(,)d Fp(on)j(eac)o(h)g(exact)h (eigen)o(v)n(alue.)212 2182 y(Obtain)d(the)h(exact)g(real)f(part)g(of)g(eac)o (h)h(eigen)o(v)n(alue.)j(These)e(expressions)g(will)c(b)q(e)150 2232 y(real,)16 b(but)g(complicated.)23 b(The)16 b(p)q(oin)o(t)g(of)f(the)i (activit)o(y)e(is)h(for)f(y)o(ou)h(to)g(realize)g(that)150 2282 y(exact)g(v)n(alues)g(for)f(the)h(ro)q(ots)g(of)g(cubic)g(equation)f (are)h(unpleasan)o(t)g(expressions)h(to)150 2332 y(w)o(ork)f(with.)27 b(Just)17 b(b)q(ecause)i(Maple)d(is)h(able)g(to)f(pro)o(vide)h(the)g(ro)q (ots)g(exactly)g(do)q(es)150 2381 y(not)d(mean)e(that)i(these)i(expressions)f (are)f(alw)o(a)o(ys)f(useful)h(or)g(simple.)212 2434 y(Finally)m(,)g(apply)i (ev)n(alf)f(directly)i(to)f(the)g(exact)h(real)f(parts)h(of)f(the)h(eigen)o (v)n(alues.)p eop bop 450 275 a Fp(72)967 b Fm(6.3)13 b(Maple)h(On)g(Line)450 391 y Fp(Compare)f(the)h(v)n(alues)g(to)g(what)g(y)o(ou)g(got)f(when)i (\015oating)e(the)h(unsimpli\014ed)f("com-)450 441 y(plex")h(v)o(ersion)g(of) f(the)h(eigen)o(v)n(alue.)450 587 y Fe(Exercise)j(3)450 665 y Fp(Change)d(one)h(of)f(the)g(en)o(tries)i(of)e(A)g(from)e(Exercise)k(2,)e (making)e(A)i(non-Hermitian.)450 714 y(Then,)g(recalculate)h(the)f(eigen)o(v) n(alues.)k(Are)d(they)f(again)f(real?)512 764 y(Mak)o(e)g(the)g(c)o(hange)g (to)f(A)h(in)f(Maple)g(b)o(y)g(creating)h(a)g(matrix)d(B)j(via)f (substitution)450 814 y(of)20 b(a)h(new)g(v)n(alue)f(in)o(to)g(the)h(matrix)e (A.)h(Reference)j(an)d(elemen)o(t)g(of)g(A,)h(sa)o(y)f(the)450 864 y(1,1-elemen)o(t,)12 b(b)o(y)h(A[1,1],)f(and)h(mak)o(e)f(a)i (substitution)g(of)f(a)g(new)h(v)n(alue)f(for)h(suc)o(h)g(an)450 914 y(elemen)o(t)d(in)o(to)f(op\(A\),)h(not)g(just)g(in)o(to)f(A.)h(Cho)q (osing)f(the)i(new)f(v)n(alue)g(to)g(b)q(e)g(a)g(\015oating)450 964 y(p)q(oin)o(t)i(n)o(um)o(b)q(er)g(will)f(mean)g(that)i Fk(eigen)o(v)m(als)d Fp(will)h(return)j(\015oating)d(p)q(oin)o(t)h(n)o(um)o (b)q(ers)450 1014 y(directly)m(.)k(Since)12 b(Exercise)g(2)f(made)f(a)h (thorough)g(study)h(of)f(the)g(complexit)o(y)f(of)g(exact)450 1063 y(calculations,)j(w)o(ork)g(n)o(umerically)f(in)i(this)g(exercise.)450 1209 y Fe(Exercise)j(4)450 1287 y Fp(F)m(or)11 b(the)g(matrix)e(A)i(of)g (Exercise)h(2,)f(compare)f(the)i(action)e(of)h(transp)q(ose)h(and)f(h)o (trans-)450 1337 y(p)q(ose.)18 b(Is)13 b(there)h(an)o(y)e(di\013erence)i(if)e (these)i(op)q(erators)f(are)g(applied)f(to)g(matrices)g(with)450 1386 y(just)i(real)g(en)o(tries?)450 1532 y Fe(Exercise)j(5)450 1610 y Fp(In)11 b(Maple,)f(let)h(A)g(b)q(e)g(a)g(real,)g(symmetric)e(3)h(x)h (3)f(matrix)f(with)h(as)h(man)o(y)e(of)h(its)h(en)o(tries)450 1660 y(as)g(p)q(ossible)g(distinct.)17 b(Obtain)11 b(B)g(=)g(I)g(+)g Fl(i)f Fp(A,)h(where)h(I)f(is)f(the)i(3)e(x)h(3)g(iden)o(tit)o(y)f(matrix)450 1716 y(and)k Fj(i)d Fp(=)600 1684 y Fg(p)p 635 1684 54 2 v 32 x(\000)p Fp(1.)18 b(Let)c(C)g(=)f(I)h(+)961 1700 y Fn(B)987 1687 y Fd(\()p Fb(\000)p Fd(1\))p 961 1707 88 2 v 997 1731 a Fi(2)1067 1716 y Fp(and)g(compute)f(b)q(oth)h Fj(C)i(C)1493 1701 y Fh(\003)1525 1716 y Fp(and)e Fj(C)1639 1701 y Fh(\003)1671 1716 y Fl(C)p Fp(.)f(Can)450 1766 y(y)o(ou)k(pro)o(v)o(e)h(that)g(what)f(y)o (ou)g(observ)o(e)i(is)f(alw)o(a)o(ys)e(true?)31 b(\(Hin)o(t:)25 b(Begin)18 b(with)f(the)450 1816 y(equalit)o(y)11 b Fl(B)i Fp(+)f Fj(B)725 1801 y Fh(\003)757 1816 y Fp(=)h(2)f(I)g(and)g(m)o(ultiply)d (b)o(y)k Fj(B)1195 1801 y Fi(\()p Fh(\000)p Fi(1\))1278 1816 y Fp(on)f(the)h(left)f(and)g(b)o(y)g([)p Fj(B)1655 1801 y Fh(\003)1674 1816 y Fp(])1686 1801 y Fi(\()p Fh(\000)p Fi(1\))1756 1816 y Fp(on)450 1866 y(the)i(righ)o(t.\))512 1916 y(Notes:)20 b(T)m(o)14 b(generate)i(a)e(random)f(symmetric)g(matrix)g(in)h(Maple,)g(add)g(the)h(pa-) 450 1966 y(rameter)f Fl(symmetric)f Fp(to)h(the)g Fk(randmatrix)d Fp(command.)512 2032 y Fn(>)582 2036 y Fo(A)21 b(:=)h(randmatrix\(3,3,s)o
(ymmet)o(ric\);)512 2114 y Fp(The)14 b(matrix)e(B)i(can)g(b)q(e)g(obtained)g (in)f(Maple)g(if)g(due)h(note)g(is)f(tak)o(en)h(of)f(Maple's)450 2164 y(usage)f(of)f(I)h(for)f(the)h(imaginary)d(unit)1042 2131 y Fg(p)p 1076 2131 54 2 v 1076 2164 a(\000)p Fp(1)j(,)f(and)h(due)g(care)h (is)e(tak)o(en)h(to)g(distinguish)450 2214 y(b)q(et)o(w)o(een)j(the)f (written)g(sym)o(b)q(ols)f(I)g(and)h Fl(i)p Fp(,)f(standing)g(resp)q(ectiv)o (ely)i(for)e(the)h(iden)o(tit)o(y)450 2263 y(matrix)e(and)i(the)g(imaginary)d (unit)1020 2231 y Fg(p)p 1054 2231 V 1054 2263 a(\000)p Fp(1.)512 2314 y(Since)j(an)g(iden)o(tit)o(y)f(matrix)f(is)i(a)f(diagonal)f(matrix)g (with)h(just)h(1's)f(on)h(the)g(diag-)450 2363 y(onal,)e(the)j Fk(diag)e Fp(command)e(can)j(b)q(e)g(used)h(to)f(create)h(an)f(iden)o(tit)o (y)m(.)512 2430 y Fn(>)582 2434 y Fo(B)21 b(:=)h(evalm\(diag\(1$3\)+)o (I*A\);)p eop bop 150 275 a Fm(6.5)13 b(Maple)h(On)g(Line)967 b(73)212 391 y Fp(The)15 b(matrix)d(C)h(lik)o(ewise)h(requires)h(use)g(of)e(an)h(iden)o (tit)o(y)f(matrix.)212 458 y Fn(>)282 462 y Fo(C)21 b(:=)h (evalm\(diag\(1$3\)+)o(inver)o(se\(B\))o(/2\);)150 656 y Fq(6.5)66 b(On)22 b(Line)150 748 y Fp(Restart)e(Maple)g(to)f(clear)h(its)f(memory)e(of) i(all)g(v)n(ariables,)g(then)i(re-initialize)d(b)o(y)150 798 y(loading)12 b(the)j Fl(linalg)p Fp(,)e Fl(plots)p Fp(,)g(and)h Fl(plotto)n(ols)f Fp(pac)o(k)n(ages.)212 864 y Fn(>)282 868 y Fo(restart;)212 934 y Fn(>)282 938 y Fo(with\(linalg\):)41 b(with\(plots\):)g(with\(plottools\):)212 1016 y Fp(The)16 b(purp)q(ose)h(of)e(this)h(exercise)h(set)g(is)e(to)g(explore)h(the)h (relationship)e(b)q(et)o(w)o(een)150 1066 y(eigen)o(v)n(alues,)e(eigen)o(v)o (ectors)i(and)f(the)h(geometry)e(of)g(quadratic)h(forms.)150 1212 y Fe(Exercise)j(1)150 1289 y Fp(Use)j(Maple's)e Fk(implicit)o(pl)o(ot)e Fp(command)g(from)h(the)i Fl(plots)g Fp(pac)o(k)n(age)f(to)h(obtain)f(a)150 1339 y(graph)11 b(of)f(the)i(ellipse)f(de\014ned)h(b)o(y)e(the)i(quadratic)f (equation)f Fj(x)1136 1324 y Fi(2)1158 1339 y Fp(+)s(4)d Fj(y)1242 1324 y Fi(2)1274 1339 y Fp(=)k(1.)17 b(Be)12 b(sure)150 1389 y(to)g(use)i(a)e(1-1)g(asp)q(ect)i(ratio)e(so)h(that)g(there)g(is)g(no)f (distortion)h(in)f(the)h(scaling.)k(Then,)150 1439 y(obtain)c(A,)h(the)h (matrix)d(of)h(the)i(quadratic)f(form)e(de\014ned)k(b)o(y)d(this)i(same)e (equation.)150 1488 y(This)g(can)g(b)q(e)h(done)f(b)o(y)g(t)o(yping)f(in)g (A,)h(b)o(y)g(clev)o(er)g(use)h(of)f(partial)f(di\013eren)o(tiation,)g(or)150 1538 y(b)o(y)h(Maple's)h Fk(hessian)d Fp(command.)k(Obtain)f(the)g(eigen)o(v) n(alues)f(and)h(eigen)o(v)o(ectors)h(of)150 1588 y(A.)212 1654 y Fn(>)282 1658 y Fo(q)21 b(:=)h(x^2)f(+)h(4*y^2)e(=)i(1;)212 1725 y Fn(>)282 1729 y Fo(implicitplot\(q,x)o(=-1..)o(1,y=-)o(1..1,)o(scali)o (ng=co)o(nstr)c(ained\);)212 1830 y Fp(By)d(insp)q(ection,)f(w)o(e)h(can)f (write)h(A)f(=)824 1772 y Ff(\024)867 1805 y Fp(1)41 b(0)867 1855 y(0)g(4)970 1772 y Ff(\025)992 1830 y Fp(.)19 b(Alternativ)o(ely)m(,)14 b(w)o(e)g(can)h(note)150 1934 y(that)c(A)g(=)h([)340 1918 y Fi(1)p 339 1925 17 2 v 339 1948 a(2)361 1934 y Fp(])386 1876 y Ff(\024)429 1909 y Fj(f)449 1915 y Fn(xx)531 1909 y Fj(f)551 1915 y Fn(xy)429 1959 y Fj(f)449 1965 y Fn(y)q(x)531 1959 y Fj(f)551 1965 y Fn(y)q(y)610 1876 y Ff(\025)632 1934 y Fp(,)g(where)g (f\(x,y\))e(is)h(the)h(left)f(hand)g(side)h(of)e(the)i(de\014ning)150 2007 y(quadratic)i(equation,)f(and)h(subscripts)h(denote)g(partial)e(deriv)n (ativ)o(es.)18 b(Th)o(us,)212 2073 y Fn(>)282 2077 y Fo(f)j(:=)h(lhs\(q\);) 212 2143 y Fn(>)282 2147 y Fo(fxx)f(:=)g(diff\(f,x,x\);)f(fxy)h(:=)g (diff\(f,x,y\);)f(fyx)h(:=)g(diff\(f,y,x\);)212 2189 y(fyy)g(:=)h (diff\(f,y,y\);)212 2291 y Fp(The)10 b(arra)o(y)f(of)g(second)h(partial)e (deriv)n(ativ)o(es)i(of)e(the)i(form)1104 2233 y Ff(\024)1146 2266 y Fj(f)1166 2272 y Fn(xx)1248 2266 y Fj(f)1268 2272 y Fn(xy)1147 2316 y Fj(f)1167 2322 y Fn(y)q(x)1249 2316 y Fj(f)1269 2322 y Fn(y)q(y)1328 2233 y Ff(\025)1359 2291 y Fp(is)f(called)150 2363 y(the)14 b(hessian)h(matrix,)c(and)j(is)g(returned)i(in)d(Maple)h(b)o(y)
f(the)i Fk(hessian)d Fp(command.)212 2430 y Fn(>)282 2434 y Fo(hessian\(g\(x,y\),[)o(x,y]\))o(;)p eop bop 450 275 a Fp(74)967 b Fm(6.5)13 b(Maple)h(On)g(Line)512 391 y Fp(Hence,)20 b(the)f(matrix)d(of)i(a)f(quadratic)h(form)f(is)h(simply)e (one-half)h(the)h(hessian)450 441 y(matrix.)e(The)e(simplest)e(w)o(a)o(y)h (to)g(get)h(the)g(1/2)f(in)o(to)f(the)i(hessian)g(is)g(b)o(y)f(including)f (it)450 491 y(in)f(the)i(quadratic)e(function.)17 b(Else,)12 b(an)g Fk(ev)m(alm)f Fp(is)h(needed)h(to)e(m)o(ultiply)e(the)j(hessian)450 541 y(matrix)g(b)o(y)i(that)g(1/2.)512 607 y Fn(>)582 611 y Fo(A)21 b(:=)h(hessian\(lhs\(q\)/2)o(,[x,y)o(]\);)512 690 y Fp(The)15 b(eigen)o(v)n(alues)e(and)h(eigen)o(v)o(ectors)h(can)f(b)q(e)h (obtained)e(b)o(y)h(use)h(of)e(the)i Fk(eigen-)450 739 y(v)o(ects)e Fp(command.)512 790 y(W)m(e)e(next)h(seek)h(to)e(relate)h(the)g(eigen)o(v)n (alues)f(and)g(eigen)o(v)o(ectors)i(to)e(the)h(lengths)g(of)450 840 y(the)h(semi-ma)r(jor)c(and)j(semi-minor)d(axes.)18 b(F)m(rom)11 b(the)h(graph,)g(these)i(axes)f(are)f(1)g(and)450 889 y(1/2)h(resp)q(ectiv)o (ely)m(.)19 b(The)13 b(eigen)o(v)n(alues)h(are)g(1)f(and)g(4,)g(resp)q(ectiv) o(ely)i(for)e(eigen)o(v)o(ectors)450 939 y(that)k(ha)o(v)o(e)f(the)h (directions)h(of)e(the)h(ellipse's)f(axes.)27 b(Hence,)19 b(the)e(scale)g (factors)g(b)o(y)450 989 y(whic)o(h)f(to)g(m)o(ultiply)d(the)k(eigen)o(v)o (ectors)g(to)f(get)g(the)h(semi-ma)r(jor)c(and)j(semi-minor)450 1039 y(axes)e(are)640 1022 y Fi(1)p 616 1029 66 2 v 616 1034 a Fh(p)p 643 1034 38 2 v 23 x Fn(\025)663 1061 y Fa(k)686 1039 y Fp(,)f(where)i(k)f(=)g(1,)f(2.)450 1186 y Fe(Exercise)k(2)450 1263 y Fp(Using)d(Maple's)g Fk(implicitp)o(l)o(ot)d Fp(command,)g(obtain)j(a) g(graph)h(of)e(the)j(function)e(de-)450 1313 y(\014ned)i(implicitly)c(b)o(y)j (the)g(quadratic)g(equation)g(2)7 b Fj(y)1278 1298 y Fi(2)1307 1313 y Fp(+)k Fj(x)c(y)k Fp(+)g Fj(x)1479 1298 y Fi(2)1510 1313 y Fp(=)j(1.)22 b(Be)16 b(sure)g(to)450 1363 y(use)h(an)f(asp)q(ect)i (ratio)d(scaled)i(to)f(1-1.)25 b(It)16 b(ma)o(y)e(also)i(b)q(e)h(edifying)e (to)h(increase)h(the)450 1412 y(n)o(um)o(b)q(er)f(of)f(p)q(oin)o(ts)i(used)g (in)f(the)h(plot)f(b)o(y)g(adding)g(the)h(parameter)f Fl(nump)n(oints)h Fp(=)450 1462 y(1000)d(to)g(the)i Fk(implici)o(tp)o(lot)11 b Fp(command.)18 b(Assign)d(the)g(plot)f(to)h(a)f(v)n(ariable)g(so)h(that)450 1512 y(it)f(can)g(b)q(e)g(re-used)h(in)f(subsequen)o(t)i(exercises.)450 1659 y Fe(Exercise)h(3)450 1736 y Fp(F)m(or)10 b(the)g(quadratic)g(equation)g (in)f(Exercise)j(2,)e(form)e(the)j(matrix)d(A)i(of)f(the)i(quadratic)450 1786 y(form)j(de\014ned)i(b)o(y)f(the)h(equation.)22 b(Use)17 b(Maple's)e Fk(hessian)e Fp(command,)g(and)i(c)o(hec)o(k)450 1836 y(the)i(result)h(b)o(y)f(taking)e(partial)h(deriv)n(ativ)o(es)h(of)f (the)i(left)e(hand)h(side)g(of)f(the)i(de\014n-)450 1886 y(ing)e(quadratic)g (function.)25 b(Obtain)16 b(the)h(eigen)o(v)n(alues)g(and)f(eigen)o(v)o (ectors)i(of)d(A)i(via)450 1936 y(Maple's)h Fk(eigen)o(v)o(ects)f Fp(command.)29 b(Extract,)21 b(and)d(giv)o(e)g(unique)h(names)f(to,)h(the)450 1985 y(eigen)o(v)n(alues)13 b(and)f(eigen)o(v)o(ectors.)19 b(Obtain)13 b(\015oating)f(p)q(oin)o(t)h(appro)o(ximatio)o(ns)e(for)h(the)450 2035 y(eigen)o(v)n(alues,)i(and)h(normalize)e(the)i(eigen)o(v)o(ectors)h(to)e (ha)o(v)o(e)h(length)g(1)f(with)g(Maple's)450 2085 y Fk(normalize)g Fp(command.)24 b(Apply)16 b(Maple's)h Fk(radsimp)d Fp(command)g(to)i(the)i (2-norm)450 2135 y(\(computed)f(via)f Fk(norm)p Fp(\))f(of)i(eac)o(h)g (normalized)e(eigen)o(v)o(ector)j(to)f(v)o(erify)f(that)h(eac)o(h)450 2185 y(indeed)12 b(has)f(length)g(1.)17 b(Use)12 b(the)g Fk(arro)o(w)f Fp(command)d(from)i(the)i Fl(plotto)n(ols)e Fp(pac)o(k)n(age)h(to)450 2234 y(sup)q(erimp)q(ose)g(the)h(normalized)e(eigen)o(v)o(ectors)i(on)f(the)h (graph)f(of)f(the)i(ellipse.)17 b(De\014ne)450 2284 y(eac)o(h)c(arro)o(w)g (separately)m(.)k(The)c(arro)o(w)g(command)d(tak)o(es)j(\014v)o(e)g (parameters:)k(a)12 b(list)h(of)450 2334 y(co)q(ordinates)i(for)f(the)h (tail,)e(here)i(the)g(origin;)e(a)h(list)f(of)h(co)q(ordinates)h(for)f(the)h (p)q(oin)o(t,)450 2384 y(here)i(the)f(en)o(tries)g(in)f(the)h(normalized)e
(eigen)o(v)o(ectors;)j(then)f(three)g(sizing)f(parame-)450 2434 y(ters)f(whic)o(h)e(exp)q(erimen)o(t)h(sho)o(ws)g(are)g(w)o(ell)f(c)o (hosen)i(as)e(.05,)g(.1,)f(and)i(.05.)k(Color)11 b(eac)o(h)p eop bop 150 275 a Fm(6.5)13 b(Maple)h(On)g(Line)967 b(75)150 391 y Fp(arro)o(w)15 b(di\013eren)o(tly)m(.)22 b(Then)15 b(use)h(the)g Fk(displa)o(y)d Fp(command)f(from)i(the)h Fl(plots)g Fp(pac)o(k)n(age)150 441 y(to)h(merge)g(the)h(graph)f(of)g(the)h(ellipse)f(and)g(the)h(t)o(w)o(o)f (arro)o(ws)g(in)o(to)g(one)h(graph.)25 b(T)m(o)150 491 y(con)o(v)o(ert)12 b(a)f(v)o(ector)h(to)f(a)f(list,)h(use)h(the)g Fk(con)o(v)o(ert)d Fp(command)g(with)i(option)f Fl(list)p Fp(.)16 b(Th)o(us,)150 541 y(if)d(the)i(eigen)o(v)o(ectors)g(are)f(named)f(V1)g(and)h(V2,)f(y)o(ou)h (migh)o(t)e(eneter)212 607 y Fn(>)282 611 y Fo(a1:=arrow\([0,0],)o(conve)o (rt\(V1)o(,list)o(\),.05)o(,.1,.)o(05,)18 b(color=green\):)212 650 y Fn(>)282 654 y Fo(a2:=arrow\([0,0],)o(conve)o(rt\(V2)o(,list)o(\),.05)o (,.1,.)o(05,)g(color=blue\):)212 720 y Fn(>)282 724 y Fo(display\([p1,a1,a)o (2],sc)o(aling)o(=cons)o(train)o(ed\);)212 802 y Fp(Prin)o(t)13 b(a)f(cop)o(y)h(of)e(this)i(\014nal)f(graph)g(of)g(the)h(ellipse)g(with)f (the)h(normalized)e(eigen-)150 852 y(v)o(ectors.)150 999 y Fe(Exercise)17 b(4)150 1076 y Fp(On)j(the)g(plot)f(prin)o(ted)h(in)f (Exercise)i(3,)g(dra)o(w)e(in)g(the)h(axes)g(determined)g(b)o(y)f(the)150 1126 y(eigen)o(v)o(ectors.)g(On)14 b(these)g(axes)g(put)f(tic)o(k)h(marks)e (ev)o(ery)i(quarter)g(unit,)e(noting)h(that)150 1176 y(eac)o(h)h(eigen)o(v)o (ector)h(is)e(one)h(unit)g(long.)j(Use)d(a)g(ruler)g(to)f(guaran)o(tee)i(the) f(accuracy)h(of)150 1225 y(y)o(our)g(tic)o(k)g(marks.)21 b(According)15 b(to)h(the)f(general)h(theory)m(,)f(the)h(ellipse)f(should)g(cross)150 1275 y(the)f(new)g(axes)g(at)f(p)q(oin)o(ts)g(whose)h(co)q(ordinates)g(in)f (the)h(system)f(determined)g(b)o(y)g(the)150 1325 y(eigen)o(v)o(ectors)i(are) 212 1375 y(\()276 1359 y Fi(1)p 253 1366 64 2 v 253 1370 a Fh(p)p 280 1370 36 2 v 23 x Fn(\025)300 1397 y Fd(1)321 1375 y Fp(,0\),)k(\()g Fg(\000)497 1359 y Fi(1)p 474 1366 64 2 v 474 1370 a Fh(p)p 501 1370 36 2 v 23 x Fn(\025)521 1397 y Fd(1)542 1375 y Fp(,0\),)g(\()686 1359 y Fi(1)p 663 1366 64 2 v 663 1370 a Fh(p)p 690 1370 36 2 v 23 x Fn(\025)710 1397 y Fd(2)731 1375 y Fp(,0\),)g(\()g Fg(\000)907 1359 y Fi(1)p 884 1366 64 2 v 884 1370 a Fh(p)p 911 1370 36 2 v 23 x Fn(\025)931 1397 y Fd(2)952 1375 y Fp(,0\),)g(where)i Fj(\025)1182 1381 y Fi(1)1220 1375 y Fp(and)e Fj(\025)1330 1381 y Fi(2)1368 1375 y Fp(are)g(the)150 1431 y(eigen)o(v)n(alues)10 b(of)g(A.)h(V)m(erify)f(this)g(b)o(y)h (estimating)e(the)i(appropriate)f(co)q(ordinates)i(from)150 1481 y(y)o(our)i(graph.)150 1627 y Fe(Exercise)j(5)150 1705 y Fp(V)m(erify)d(in)h(general)g(that)f(the)i(scale)f(factors)896 1688 y Fi(1)p 871 1695 66 2 v 871 1699 a Fh(p)p 899 1699 38 2 v 899 1722 a Fn(\025)919 1726 y Fa(k)941 1705 y Fp(,)g(k)g(=)g(1,)f(2,)g (indeed)i(con)o(v)o(ert)f(nor-)150 1760 y(malized)e(eigen)o(v)o(ectors)i(in)o (to)f(v)o(ectors)h(of)f(precisely)h(the)g(length)f(of)g(the)h(semi-ma)r(jor) 150 1810 y(and)h(semi-minor)e(axes.)27 b(Begin)16 b(b)o(y)h(obtaining)e (\015oating)h(p)q(oin)o(t)g(v)n(alues)g(of)1403 1794 y Fi(1)p 1379 1801 66 2 v 1379 1805 a Fh(p)p 1406 1805 38 2 v 23 x Fn(\025)1426 1832 y Fa(k)1449 1810 y Fp(,)h(k)150 1865 y(=)d(1,)f(2.)k(Then)d(scale)h(the) f(normalized)e(eigen)o(v)o(ectors)j(b)o(y)e(these)i(factors.)j(Next;)c(use) 150 1915 y(Maple's)e(arro)o(w)g(command)e(to)i(build)f(a)h(plot)g(of)f(the)i (ellipse,)f(and)g(the)h(newly)f(scaled)150 1965 y(eigen)o(v)o(ectors.)19 b(A)12 b(plot)f(of)h(the)h(ellipse,)f(and)g(these)h(new)g(basis)f(v)o(ectors) h(should)f(sho)o(w)150 2015 y(that)18 b(with)f(this)h(scaling)f(the)i(v)o (ectors)f(coincide)g(precisely)h(with)f(the)g(semi-ma)r(jor)150 2065 y(and)c(semi-mino)o(r)e(axes.)150 2211 y Fe(Exercise)17 b(6)150 2289 y Fp(Find)f(the)h(equation)f(of)f(an)h(ellipse,)h(cen)o(tered)h (at)e(the)h(origin,)e(and)h(for)g(whic)o(h)g(the)150 2338 y(ma)r(jor)c(axis)i
(is)g(4)g(units)g(long)g(and)g(lies)g(along)f(the)i(line)e(determined)h(b)o (y)g(the)h(v)o(ector)150 2411 y Fj(v)170 2417 y Fi(1)189 2411 y Fp(=)239 2352 y Ff(\024)282 2386 y Fp(3)282 2435 y(4)323 2352 y Ff(\025)345 2411 y Fp(,)k(and)f(for)g(whic)o(h)g(the)h(minor)e(axis)g (is)h(2)g(units)h(long.)30 b(\(Hin)o(t:)d(If)17 b(y)o(ou)p eop bop 450 275 a Fp(76)967 b Fm(6.5)13 b(Maple)h(On)g(Line)450 391 y Fp(can)f(\014gure)h(out)e(the)i(eigen)o(v)n(alues)f(and)f(eigen)o(v)o (ectors,)i(y)o(ou)f(then)g(can)g(\014nd)g(matrices)450 441 y(Q)18 b(and)f(D)g(for)g(whic)o(h)h Fj(A)g Fp(=)f Fj(Q)7 b(D)i(Q)1037 426 y Fn(t)1051 441 y Fp(,)18 b(where)h(A)e(is)g(the)i(\(symmetric\))d (matrix)f(for)450 491 y(the)k(quadratic)g(form)d(corresp)q(onding)k(to)e(the) h(ellipse.\))32 b(Graph)19 b(the)g(ellipse)f(and)450 541 y(the)c(scaled)g (eigen)o(v)o(ectors)g(for)f(A,)g(demonstrating)f(that)i(the)g(prop)q(er)g (scaling)f(of)f(the)450 591 y(eigen)o(v)o(ectors)h(giv)o(es)e(them)g(the)h (lengths)g(of)f(the)h(semi-ma)r(jor)c(and)k(semi-mi)o(nor)d(axes.)512 640 y(Since)18 b(the)g(axes)f(of)g(the)h(ellipse)f(are)g(orthogonal,)g(y)o (ou)g(need)h(to)f(get)g(a)g(v)o(ector)450 690 y(orthogonal)10 b(to)h(the)h(v)o(ector)g Fj(v)914 696 y Fi(1)933 690 y Fp(.)17 b(This)11 b(can)g(b)q(e)h(done)g(in)f(the)g(plane)g(b)o(y)g(in)o(terc)o (hanging)450 740 y(the)21 b(x-)f(and)g(y-co)q(ordinates,)i(and)f(negating)f (one)g(comp)q(onen)o(t)g(of)g(the)h(resulting)450 790 y(v)o(ector.)e(Ev)o(en) 14 b(a)g(casual)g(insp)q(ection)g(rev)o(eals)g(wh)o(y)g(this)g(w)o(orks.)p eop end userdict /end-hook known{end-hook}if