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

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

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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 =-. 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