2025pr5 - Math 2025 Quiz #3 (Fall 2004) Name: Inner product...

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Unformatted text preview: Math 2025 Quiz #3 (Fall 2004) Name: Inner product on functions on [o1 b] (piecwise continuous, continous, etc.) is given by” (f, g) :- f: f(t)g{t) cit. The norm is b an¢onm ff@%e If A is a Subset of R“ then the indicator function of A is the function XA : R” -+ given by 1 if t E A {A 2 X“) {a if z: e? A In the following we will aiways use a 2 G and b = l. A) Evahlate the foilowing inner products: a W l fiHflrLflflfimfi: g ;ngggtnwf3 (2) Let ft‘b‘) 1": t2 and 90:5) :5 75 ‘t’ 1- Then (f: 9) 3 ; (3) Lat fa) E Hmong) and 905) E X[o,1/4) (if) * XEl/e,i/2)(t)' What iS {fig} l“; (3 B) Determine which of the following pair of vectors are orthogonal. <1) (1:W132)>(1311“1)3 NC} (2) fit) Sin(27rt) and 9(t) : cos{27rt); Veg: (3) fit) 2: t2 —~ 1 and 9&5) n t; We H C) Evaluate the following norms: . gmw‘” nmm:”t2 (‘2) “(22334)” 2 n“; ; emwwggxeag (1) i &E Q (UV-Him WAN}: \W" 2% '1"; 3’4 ($13): SERwQCM w a “a A - “" Sgt, #"LMW «Ewimw? V1 3"“ @5 - 1 W mu ¥\3\) W w {a} lwr‘é; "’ a ’ \ 1% I >4; L055“ 133; m‘zELQM wiwwum;flm¢wi 2% ‘1 “3:0 3mm; Q} a) (mmm ,3 i h NW3: MSQM-qfil‘ gmgqu ' EW' \dfi QM «Ci/2;;anth mi“ 53 79M (3Uch :1 a my 2W (1 ‘ fi‘wai’iflggmaflfl‘a vugfifi.%3M 3/) Sigwflhfiv _. -‘ :5 , a, K Mg .. ,_ * ,. a t 3-: _ 3 Mi Wag/ii ‘1 E 3: C > A) ncsffifi mac 02 3g 75) May-0112: wquamm ‘3) :x&e§x\l$, g‘wkz u; i i g 2 “Q “yikzefl 3+. 2* , w «- 6‘” ~® M m “7- ” 1 E I '34» “m fitg t3 4-1 ez‘tg 5 ~ E 2 5 0 .... m 6&3} 1-: g _ Z 'a fiLewfl <33“... 1 ag+q Math 2025, Quiz #3 (Fan 2003) Name: SoMQfihs Reoali that an inner product on R” is'given by ((321, . . . ,Lvn), (yl, . . . 4:71)) 2 my; + . . _ + ssnyn. The length or norm of a vector u 2: (331, . . . ,m) is the real number %’($l=---a$nmfl \/$§+$§+-~+33i= Wm) Inner product on functiono on [m bf (piecwise continuous: oontinous, etc.) is given by (f> g) n f: f(t)g(t) din The norm is b m z m: 1) anlua’ce the foiiowing inner products: a) 11 x (1.,2ju—1),v :(—1,1,1), (11302 O‘ m.— m (WU): l-C—fl+2-E+C-D'i= -l+2-1:-a b)(L:{),b:1,‘f(t).xfi2+ljandg(t)2mt+2' (fag): jag/{Z £C€x}5£fi):C-£L+l)(2 H‘s} :2£2+2*~€3-_t, (€13):- Slatzuklmtg... tau ,,._ . ' a o :«ft3+2tm~,iifi~§1ang~+2m§i mi , gum/“24: fl: 2.3/72 /2 2) Evaluate the noun of the following vectors: a) u : (1,»—2,2,1), = /éfi7 ixullrmr§i+w7731/m b) a 1: 0,35 : 1,1" Magi/2) “ X[1/2,l): 2 I 2. "1m , r 2 £5“ ‘7 View“) " “mu/o“ 7‘ 2:21) C” “(Lea 1)“) = We 1) ‘9) 3) Which of the following ‘ i1; Mmys are orthogonal?‘) g; wig/«)6: .1: 1 $94 a: {:3 1; ’ a) 11:: (1,—1), v m: (1,1). i ‘r’ (LA M = t ~g : o ' b) omU,b~—:1,f(i)xtandg(f)=2m—3t. fi‘reg I £66306.) ~ élfiwgta‘ k , 2 2 3 ."2'" l «.l a: c) (fia)-— gal-bsgtdbfim fiwt a Math 2025, Homework 3, due Thursday, Nov 14, 9.10 AM Name: Consider the vector Space V of real valued piecewise continuous functions on {0,1) with the inner product < fig >: f; f{t)g(t) dt. Lot 90%) 2 90(22: — j) as usuaiéy. Let for) 2 t2, g{t) : t, and Mt) : 2t — 3t? 1) Evaluate the inner products: 8») < fyg >3 b) < 9,903 >3 c) < 9, 90% >3 d} < f,h >= e) < hufi >2 2) F End the norm of the following functions: a) m z 10) HQH = C) M” = d) W e 3) Consider the subspace V1 2 {30993 +8190§ E V f 30,51 6 R}. Let 1 , 1 : 1:510:90? (éigl‘f‘pi ‘ a) Draw the graph of p and g. 1)) Evaluate 339 -p\ 2 c) Show that if q 6 V1 then 9 mp i q. 19 Math 2025, Problems / Inner products Recall that an inner product on R” is given by < (51:1,, . . . ,zrn), (3,11, . . . ,yn) >: 221311 + . . . + xnyn. The length or norm of a vector E) 2 (3:1,. . . )an) is the real number H(:z:1,...,:r:n)ll:«/$§+3:§+...+33%: V<3€fi§> 1) Evaluate the inner product of the following vectors in lit”: a) < (1,e,-—3),(2,1,3) >x1><2+0>< 1m3><3==2~u92w7 b) <(2,2;w1,-~1)?(Z,1,2,2) >22>< i+2><1—1>< 2e—1>< 2:0 c)<{1,273,4,5),(0,2i5,3,6)>21><G+2><2+3><5+4><3+5><Gzfii 2) Find the norm of the following vectors: a) “(1,3,1)He 2 «n: 3. 317 b) ||(w1,2,3?5)n 2 war)? +22 +32 +52 2 M39 2 6.245 Two non—zero vectors are called perpendicular or orthogonal if the inner product < "13?, m3"? >= 0. 3) Which of the following vectors are perpendicular to each other? (1:0)? (0 — 2). Orthogonal b) (110,1), (~l,2, 1) Orthogonal c) (2, ~l.70), (l, l, 3) Not orthognal (inner product = 2 m l = 1) An inner product on C” is given by < (21, . . . ,3”), (11)], . . . awn) >2 2110—1 + ~1- znwn where :17 — 13y. The norm of “z? m (21,...,zn) is given by denotes complex conjugation n+1 < . y 3 = x/l21[2+...+lzn!2 = “l E? Z >. Two non—zero vectors E} and 3? are perpenticuiar or orthogonal if < Mi"), “57) >: 0. 4) Evaluate the inner product of the following vectors in C”: a) < (1,i+2‘)2i)5(—1,1—332+3’) >: 1><(—1)+(1+3')><(1 —1:)+21;><(2 +1) : —1+1+22‘+2r+2 m 2+43‘ b)<(i123,1+4i),{1m1;,3mi,3) >=r><(1—-r)+2r><(3 —-i)+—(1+41')><3 mé—1~%~6tm2+3+12i: 2193' 5) Which of the folowing two vectors are orthogonal to each other: a) (1,1531), (1,1+?§,1) : 1x1+1><(1+3)m3x1=1+1711~1§= 2w2i not orthogonal: b) (1+«i,i, 1—21), (1—3; 1, —1—r): (1+r)><(1 m 1L)+i-><1+(Zwi)><m =1+2iw1+3w2 :3 W2+3i not orthognal 6) Find the norm of the following vectors in C”: a) “(1 +122 — 375i)“ w W 2 4.0 b)||(t,1,—1)llm mm «$21.73? WLet f : [on b] me C be a continuous function. Then f can be written as f : ' ML 11 and 11 are continuous real valued functions on [(1, bl Then integral of f over [a: b] is then defined J. ’L’) + tote) where Eras fabflw) d3: : (£3: +i/j dx ...
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This note was uploaded on 11/23/2011 for the course MATH 2025 taught by Professor Staff during the Spring '08 term at LSU.

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2025pr5 - Math 2025 Quiz #3 (Fall 2004) Name: Inner product...

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