MATH
full-solns-problems-3-223-F07

# full-solns-problems-3-223-F07 - ?LD€L€M SET 3 1...

• Homework Help
• 10

This preview shows pages 1–10. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This preview has intentionally blurred sections. Sign up to view the full version.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ?LD€L€M SET 3 1 Soeuﬂoms. 3046 WWW; WWW/2 (gr r6575 ask a+ recVFmL/WID Q‘ié. Suppose WW’IL U“ max 1:; are “I'm/D Ingmlvrs , 5‘1 41/2 ofﬁzuoer-Q bask ‘71? R3- W9 Wm‘ +0 show kiwi We, WW Wd?!i,,’f§e¥'a‘?ixr‘} W63 Maw HMZL ’V' 00f W Spam/L @ 2 dimems/ma/g 7 éuks/QZCQ, (L R Thu/y ‘HAW (is co (L— p6/m6MSImﬂ/e Suéapace ﬁ/(Z _ wuver'DG £5 IVlOi/lcbt/Qa/k 4v- 7))“ wave (715‘ MmW/ . ., (73x55 7 aUchom 145 \$33 [fr 5,0an ueadibr‘ H in “HAL 5(0qu Vém/pinaﬂwwgm ‘(0 3‘2” J , WW 3’; V'X W J— % ﬁﬂ/WQMA/ea/L, +5u; 7'7!)in M\$ (JV/(Z 'tdw, mmmwe boa/44A W‘i ire a W 6{» U'KW' we [bu/1014f W {ﬁrm}! 5 W00 0’36 [pa/LOWXA’W ﬁﬂam/Nlai éjv‘ﬁw (W = {C ﬁr“?! Writer ,777 r e, 17 M Wee.) ,, Heme - ,e. M ,, XMFneol. ¥_ ,, .. 2,. 4.; (a) 11866 ’FOUF Veci‘org 0346 Vié: --.-777. 7-, .-.,. . Uneqrbr.?od€iem499i1wf%r7qmgi237- _ ii: [i ii: 1 + ++ F W _ u ” A[3.71:1ffiiiaﬂfﬁ: ~—[:7%:}[:g EC .. 7 7. 77;. (7/. ES ,0 (.17.; (3/395 . Pa .14 n t: A . 7 -7 L/ 7 7 7 The \$767. .7... 77V579£97C§7777 (S77 7. 7.7.-- 7©Jringram7a Mac-775411; 50m 31/2510an ./ 7.0(6/pci/71 cling m7. .7 7 i=0 wz7tip7777--77----777777- 7 7.77... 7,.- .m-._7_77.__‘-..‘7 ... ...~_~_7777 .7..- 77.7 77 77. 77.77.7........7..~-7_.7.7_.. ... -777V—t F77.-. wla 71.2;(47. 7 7 57%.5 page. 7.749% a ((7.... 7114479770047 7 7. . if,“ Form 7 1:27:77. -/. . w7/Fiﬁ. 7. ..-a/bga R77. 77- 7 2523Lairijwth—QRWLWAM _777f5/_-e °ricai5€_f°£_EO--7777tmpijes_777Im5Cj)=i€WL7C3C7>7_-_777 .7777F7A L S E 3-77.77977907757EEC7..7._-77__7:£.E._.Q7777-M)\777.7877?..5677777777----.Ziiia.€77-74: 67m7z77£m Lifig nS77... Q 7777777Th€ﬂ777f°g;o -_ 75 -77 176%., .7 Edi. IMj£j7iESLaj_-OJV\QQ___77 -WmmwﬁmﬁLikﬂm7 777 .--- ..... 7 ........................................................ “"‘”‘""“""””””“S"A”L‘VA‘QE : ”W M, 75‘ #56 ”is" i:(’\ (1+ ' " ”‘ ' ' "‘ '" 7.7--. """ W1 MATH 223 PROBLEM SET 3 , ' SOLUTIONS 1. Prove one of the following statements. (a) If (V1, . , . ,vn) spans V, then so does the list (V1 — V2,V2 ~ V3, - . . ,vn—1 — vmvn) obtained by subtracting from each vector (except the last) the following vector. PROOF. We need to show that for any v E V, we can write v as a linear combination of (v1 ._ V2, vz — v3, . . . ,vn_1 — V“, V“). We know that (w, . . . ,vn) spans V, so we may write (1) v : am + - - ' +‘anvn. We are looking for coefficients X1,Xz, . ,xn such that v = X1(V1—V2)+X2(V2—VS)+'~+XnVn. Let’s set X] — (11 X2 — C12 + 01 X3 — as + Cl2 + CH xn : an+an_1+-~+a1+a1. Then the linear combination x : x1021 — V2) + X2(V2 — V3) + ~ ' ' + xnvn becomes x = al(V1_V2)+(a2“al)(V2—V3)+"‘+(an'—an—1)Vn = GM—C11V2-l-(dz-l-Clilvz—((12+C11)V3+-~'—(an—1+~~+aiivn+lan+m+a1ivn (1]V]+"‘+C111VTL V = v by equation (1). Thus, this linear combination is v, and so this list spans all of V. El (b) If (V1, ~ . . ,vn) is linearly independent in V, then so is the list (V1 — V2, V2 ~ V3, . . . ,vn_1 —~ v“, V“) obtained by subtracting from each vector (except the last) the following vector. PROOF. To prove that the list is independent, we must take a linear combination adding 11p to 0, and prove that all the coefficients are zero. 80 suppose 0 = bilvi —-V2) + b2("2 —V3) + ' ' ' + anw Then we may rearrange the right hand side to get 0 : blvl — MW ‘1' bZVZ — bZVB “l” ' 'p' "l‘ bn—an—l _ bn—lvn "l’ bnvn = b1V1-Hbz — b1W2 + ' ' ' + (b'n — 1311—1)an But this is a linear combination of the list (v1, . . . ,vn), which we have assumed is linearly independent. Thus, the coefficients of this linear combination must all be zero. In other MATH 223 PROBLEM SET 3 SOLUTIONS words, by = 0 b2~b1 _ o b3-b2 —- 0 bn "“ th—l : 0. In other words, b1 : 0, b2 2 b1, b3 2 b2, and so on until we conclude that bn : bn_1 : = b1 : 0. Thus, we have shown" that the list (v1 — vz,v2 —— V3,. ..,vn_1 — umvn) is linearly independent. [I (c) Suppose that (w, . . .vn) is linearly independent in V, and w E V. If the list (w + w, . . . ,vn —I— w) is linearly dependent, then M) E span(v1, . . . ,vn). PROOF. By assumption, the list (w + w, . , . ,vn + w) is linearly dependent, which means that there exist scalars a1, . . . , an E R not all zero so that 0 : CL1(V1+W)+C12(V2+W)+“'+ (Ian—I-W) = mw+aWys~+awW+wvs~+aaw. Because the list (v1 , . . . ,an is linearly independent, the coefficient on w must be non~zero,. In other words, a1 + - ' ' + an 7; 0. Thus, we can rearrange the equation to get —Mrs~+adw=ast~+awm and since —(a1—|—-~+ an) 75 O, we can divide by—(cn + ~ ~ . + an) 7é Oto get _—g]_—v .I. ..+—___EL_T"— —(al+"'+an)l _(al+"'+an) In other words, w is a linear combination of (v1, . . . ,vn), so W 6—: sp an(v1, . . . ,vn), which is what we wanted to show. [:1 W : V“. 2. Find a function f : R2 ~+ R that satisfies f(a - v) = a . f(v) for every a E R and v E R2, but which is not linear. EXAMPLE. Consider the function ' mm) = v x3 +133 Then “CL. (x,y)) = «3/ a3x3 + a3y3 : av \3/x3 + y3, so this satisfies homogeneity. However, “(0’2” : {/51 “(290” : {7—8—1 bUt “(0:2) + (2)0» : f((2>2)) : W 7é ‘73 + % Thus! this function does not satisfy the additivity property, and hence is not linear. <> 3. For V and W subspaces of IR“, pfOVe 3that (a) V D Wis a subspace of R“; and . (b) V L34 W is a subspace of Rniif and pnly if V g W or W g V. (a) To Show +l’\0‘r+ V/{W L5 0! 53U!95,)Dac2./.3 (Ale maecé +0 ‘ show 1")“: closeJl 3 Dumber sax/Cam. mulﬁplicadiw mot meats-c AMf'h‘Q’Vl r 3 3 3 3 3 3 3:333:36 suppose ire VOW ‘ Mal ja. 6—02. Vela U‘év Wok V‘C'W 1' 427 aﬁeﬁnlﬁw 9f "4+8V‘Se chon/ @914 571463, ﬁws‘e (arej subs/Jaw, 6: (rev myf a ()th ' BuiL W W Exao'HJ “3994+ awrt‘Vrﬂ/U'. Segonﬂl If: WW9VOW W 307we~v MA , WW6 W Shea m6; m ‘ gutlosfacuj V+wev CW v3+w arevl oi (++u)+c1r2~ch («SM‘fraaIr/‘I‘m 3 031/121 cape/Ce, ' 9(6 wa 3:14 ”((4231 Mound ‘ Ufﬂu‘ FWeW / So (uwﬂéw) U‘GJ/ FEW-{Ta c0133 3W1 MOICZ éfr U' 'W/ 4erQ/t'hmfl/ 3 M I; A497L 43 6ubsfa6ce, l 4. We say that two subspaces V and W are orthogonal to each other if v vw : O for l ‘ every v E V and w E W. Given an m X n matrix A, let R(A) denote the subspace of JR“ spanned by the rows of A (thought of, of course, as column vectors!). Prove that RM) is orthogonal to the kernel of A. Deduce that the image of A is orthogonal to ”the kernel of AT, , , , ~ , , RM) 1. lap/\(JAY): For guru veal—or We’lﬁMlJbD) ‘ A'U’Dé 130+ y “WAD. :01" moralinm {{L A“? is ‘l’la/m oaela MOOAA\$4 ‘ {Cub 1% 49v)” 6? A: balm FL Hot/(ca, MM fﬂw 040% News m o, w t a gamma/3+0 3 M W ﬁg 74/ Mal Wee +79 stC/LawSUQlZPM. .MWwaw‘3 *****3 : we» Mamas (a mag 1mm: CKMF‘RMT), tamer/x IMCA) ,1. zen U17“)? 4,? meat fart. * f” * 75. For subspaces V and W in IR“, define their sum to be : V+W={v+wlv€V, WEW}. (a) Prove that V + W is a subspace of R“. b What is V + V? ((c; Prove or give a counterexample to the following statement: If V1 , V2, W Q R“ are subspaces such that V1 + W = V2 + W then V1 2 V2. ‘ ‘ ‘ (d) What can you say about dim(Vl1‘ + V2)? l l l I; l 5(a); V+W rs Vol/052% WWW (rec/+177, gwﬁ‘w/ sit/10L :(m'l'l'VQtB l7 Lu; ”ﬁx/017A :l (1)11” 0.7,” + tw¢+wzj stings :2 what; (one); ‘31M 3: 1 W M were» ‘ warm : 51C) ‘ We 9T0L+€VY1€W+ IS F71} LSE For txamFleJ 't E 9.4 V- i ﬂ 1 ' V1: {(03) €1l<3”{ « \/ g {(25) .9 7’3 ; ou'apois 6. Let T : R“ —> R“ be a linear transformation. (a) Suppose V1) , x72) 6 R“ are non—zero, and satisfy T071) ) : cm? and T073) 2 @273 for or] and ocz distinct real numbers. Show that {W , \$3} is an independent set. (b) Suppose T2? , . . . ,Wk 6 R“ are non-zero, and satisfy TH?) = out)? for 1 g i g k, with the oci distinct real numbers. Show that {W , . . . , 97:} is an inde endent set. (c) Exhibit a linear transformation T ; R2 ——> R2 and find a basis {\7’1 , v2} of R2 with ' W1): 3V1) and T073) '= 5‘72. ' (d) Exhibit a linear transformation T : R2 ———> R2 such that for no pair of real numbers (om, ocz) is there a basis {V1}, Q} of R2 With Th?) ‘= cm? and Th7?) = (X253. gLQ) We Whom \H/Ltts l0] FHCJUO7L7\57/| 6% )6. POM/7L 5(a) ‘Provt‘oif’é m (ﬂasﬁ cage. (ASSumeﬁi UT, ) WV [\$-11 W 1 WWW; Mid {V 5%? '64 , ~~ y U]: 7—9 W- ‘ , ‘ .3 ~, ‘4 ‘ ‘ ‘ . 1 7 : ‘ : ‘ Vic: L; Cc U"; (Q is a 61mm Gawain/maﬁa” ‘ ‘ ‘ ‘— 5' VI; 3% Wm 4 : Rig ‘GW’IC‘PJQZBWFJ c; ‘ me. ‘ 444774“ «Ii (9; é/Vngg é’:{=5_ ‘WJ/Ej'nj T, WC («W ‘ ’6‘4‘ J 775); ﬂag/7cm) 1 k—l ‘ ._ __._l p A ' ' ‘ 5 lg—I ' ‘ lg? ' ? g) = "5 :. C‘O(~ (j‘ is sulo‘h‘i‘v'l? >09(@23IC1U~¢) (:1 ‘ K 1‘: L J _ ‘9 k—1 : ‘ ‘ J ‘ ‘ 1 =7 0 =-. Oszw/UUQ 1;/ ‘ ‘ 3 ‘ 1 I .‘ I ‘ ‘4 i 'TLW5/ 1W6 “Km 4 264% Cé’Vl/Lblkﬂ/‘EW {ﬂ 0;?)‘H/ka’l /‘ elmz ,W on”; Q W :ikﬂhfj‘- 7741‘5‘ mud‘MdM: MJVOMﬁS“ ﬂage— V7177 Vic-1! m Mate/2.0mm iww Icicjanéf M O i W 91"“30: #0 k'w‘i) ”M3 = (:3 Tb SouLIs‘Exj , Tug): «(75) BU+ There is iii) 10W 9/; [221 wind/1 haw “HM; yvcoord/L‘tht 0 ha Aofél3 .Aa/J/{S V‘eV'é‘DPS. (we, mew «Mow H W; WM 50on (mfg W4 347”}, 501%; n/wnva/wo; . mg wow—L2 AKA ...
View Full Document

• Spring '08
• TARAHOLM

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern