Math 33A Winter 2008 Midterm II

Math 33A Winter 2008 Midterm II - UCLA Math 33A Lee...

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Unformatted text preview: UCLA Math 33A Lee 2, \Winter 2008 Midterm 2 Feb 29, 2008 Last name: First and middies Names: UCLA ID number: Section (circle one): 2A WONG, WANSHUN Tuesday 2B WONG, WANSHUN Thurséay 2C KWON, SOONSIK Tuesciay 2E) KWON, SGONSIK Thursday Instructions: 0 Fiﬁ in your name and eircie your section above. C You are aléowed to use your brain7 pencil/pen, eraser only. Calculators are not aélowed‘ c ‘Wz‘ite your soéutious on the midterm papers as possible as you can. 9 Show all the necessary steps invoived in ﬁnding your soiutions, exeept True—False questions. Problem Points score 1 10 2 a 7 3 i 8 4 7 '5 8 Tom} 40 I Problem 1: {10 points, 2 points each) Indicate whether each statemth is true or false (you need not justify your answer here). 1. R2 is a subspace of R3. g l; x a _ i f “2 3 . i ‘? ’a < “1' , ‘2 ix: x ,w ' iii a e g éiiﬁr ’ a m a? r3"? m: e, a ﬁe“ er a; a g . V - f i 2. If A iaawmetric (Le. AT m- A) and invertible matrix, then A” must be symmetric as well, g g E _ Hi , ﬁé’wﬂi ﬁg» 3: 5 5% Wii a? mi '9’: 5 :5 m g a g . 1,. ' f ‘ " ' _ {’3 3. Ii Eli is a matrix, then there exist. matrices Q and R such that M m QR, Q has columns which are orthonormaiigi each other and R is an upper triangular matrix with positive entries on the diagonal. g g: ‘ i .‘M‘t ; ‘ i i » ’ , I i f = “a. j ' 2’ " *‘“ Haik‘. i l, - ; , . t as“: "a. e , t, .w’ ,2“ ﬂ g; tag, tr: “getter-rig} were ix? ﬁre a ﬂew a"? {ii We at; E 4. If T : R” +—> R“ is an invertible linearlgransformation, \$8 is a, b. " he R”, anti B is the matrix of T with reaped; to the basis %, thenB must be invertibleg : . = ‘ _ ,7 we? 1 is? 3 “d? w“ "'L M ‘ a, {6:} (a ,- i V‘ ’ r «3w m: Q r" ’1’ 5W} x" r m - or w ewré’i‘éar i “riﬂe: ﬂirt-egg: r‘i ( SW? i5 E’ékgﬁ ,, fiiei‘égg’sﬁ Waite??? ﬁg; U 3 5' '- m x f k' ’ W K _ i k r a @313 if; E; gage: r“ giﬁie \$31” iikaéﬁ ii} Efﬁe/rel 7:: g”; .5. if W’ and V are srlbep R” and W g V lie, W is a subsetfof V), thenxit must also be“ Problem 2: (7 pointsﬂeé: A be the matrix (a) (4 points) Find a basis for ke:‘(A). (b) (3 points) Find a basis for Im(A). if WE wzﬁaﬂgﬁaﬁiﬁ \$5" { s g a g 5? L; if? Mégé’ﬁ} if a :2 a * é ( ‘ I A W g3 f \$5; “\$51k? 35% :: €§§WE a: ' a - * i gag Tim a»: a éig‘zciéﬁgaa 5 K 5 E a}? 1_ x ) be the matrix Wm; 1’1 3 ’U2 , "U3 m; Problem 3: (8 points) Let A = ( 51 my gay Wm . 1m 3 .m WEE§§E a i. m 38 £5; % . (as? a; 3%, m mﬁgﬁg; {W mg a O , I. Ham 9 5 Wm, any.) .23}; £1 C aim 5...: O MWM/‘muws A; g Lug «NU “va , 3 1% m 1% gig a , , J , _ _ . . . ., {UV 2 a w Egg 3 § § 3 5a 2 .m e E?“ ii; ; 2: m a? ﬁg M} g : x g 1m 0 § " my K g, Ewg t a m fa r? W 2 %§§§%,,; g, R o m : Egg; 3 a my a», . Q m a. Q 2 3 {x V2,, \ull/ M «Q m3 3.39%? E§§§ xmf NW; \$5,333: 23?; 332?} M, 0 2 2 «G .mdéfhﬁ v, 3 Z ﬂu Zzﬂw, z gﬁl as aw mg?” 2 O O n “\$455311 Wig» ﬁjmfzaunwr, XEVﬂgx. S w iw a £53.13? is; w ﬂ is; Ea}; gaziisi V x 034 m m siﬂmuuﬂiai m: f 3 f ix m m c A, 3 W a% a; «w a MA} ‘ ,. r m 2 BE; . meg/«y 9mg? 3, s W x, 5.2.3.5: EL”. “a. M p O a. 3. Egg. My «Lu §§a 3;» t e E: __ I KNEW .2? i, 1; g 3 j m m ﬁg ﬁwwﬁ ad; i g M Wm 33 f x; E, £5 1 h o f. mm .m M : : : : : g : 1 1 sx .xw wna Mm 523.32: M w is my a}? by 3%}; WWI/Av 5.1% mm m :1 am 3%? Pay A in“ 3/” W aﬁsm a; i 3?: “My 8 t ,3 , is A: 4,?“ 3%,, K he m M? a: y: a ‘ t d a w .m éaé U F a, \s) Q m. 7m . .m am mm. : p p d w Q m ﬁn. a :0 {3% 3 If Ag? WW Let T be the linear iraasformatéon from RE to R2 deﬁned as ) Probiem 4: (7 points _ 1 2 _ 4 3 . (a) (5 points) Finé a. basis 53 of R2 such that the iB—matrix of the linear transfermation T is where A Problem 5: (8 pail/ﬁes) (4 pants) Consider an n X m matrix A such that ATA m 1m. 15 it necessariiy true that AAT m In? Explain. (Hint: consider two cases: 71 = m and n :34 m). (b) {4 points) Is there an orthogonal transformation T from to R3 such that TE) = (a M?) = (\$3) if so ﬁnd one: if no: explain Why not. 5:} 3? ifé 21%??? ﬁfiiém 5% i}? 5% mwéié éaé’ é‘ﬁ QM? 2% :- if “é'im 5‘ E a ; kg W. «pg j M 1/5 {a}; 53 R xii: {Ema E2251 yaw? gig-“a \$3? \$3? {if/mm}? :i‘. g 3 x . w% w 35:1.i%: 7“‘; 3% “gm”? 54?, N‘Eéﬁw Mﬁfgﬁg @5453? E .J 5;? 3 9- ., "f; , if ,«gwr' ' {égws :EQZE‘M) 5%m1 ﬁwiééﬁa; r:- \$5; ‘3; ﬁ®imww mﬂﬁﬁﬁm<aiwmkmwmmwwag @Eﬁ ’j r g E § 37 '~ g a g .K' ‘ __ _, f_ féaﬁxé‘ééiw ;% £3: aim—é 52%: 3;? ...
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