MD term 2 - ., n.»— .WA '(30 pts.) 3. Let V be the...

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Unformatted text preview: ., n.»— .WA '(30 pts.) 3. Let V be the vector space of 3 X 3 matrices. Let A and C be two nonzero 3 X 3 matrices. Let W={BEV|AB=O} and U={BEV|AB=C} a) Show‘that W is a subspace of V. A , C) :L O b) Show that U is not a subspace of V. _ l;\ e (“u r; 1:: EU AuozfliC LSW’ Sc. o¢LA .9; ti. 0) Determine W when A is invertible. Let gene, “We. Llama; A'QQZO Q): ESQ: PNQAEl: AV‘020. Q‘ljog b Shee- (1) Determine the dimension of W when the rank of A is l. (D g: Kid Sf“ CC MAL 0(— ‘g R: 1' A . (3 ts c, L W! Wee-id!“ E f O O W O O O 1 S O r l“ is 7‘ 1— ‘“ 7: f KL “‘3 x g \-: 1, 7‘; ya“ 3% X} 7%, \IWEVU 93$ .— 0\lL'vllollé'l/‘l' («X to \ but) 5 \rfeo— Aka «Slur a l" l'lflé Cclui‘wA l’ éiMQM) ié ,1: M E T U Department of Mathematics Basic Linear Algebra Midterm II COde : Math 260 Last Name I Acad. Year: 2008 _ Semester 1 Fall Name StUdent N0- Instructor 2 Department: Signatiire Date : 24.12.2008 Time 2 Duration 1 .90 inc 3 Questions on 3 Pages Total 100 Points Show your work! Partial credits will not be given for correct answers if they are not justified. (30 pts.) 1. Let 0 3 —3 1 —5 A: 01 ~1 0 ~2 0 1 —1 1 —1 3) Find a basis for the row space of A. O \ 4; o J}. t»; +2.? ‘0 0 O l l .hllli‘fll} ' 7’ “1‘ a - e o o \ A .——"/° A (D \ -1 Q .-’L 4 I -1) i7\_‘i‘ \‘- _ Q 7 l L C} O 0 l l plaid. O O O O { Lo \ .-1_ O -11 {mocill} i3 ox LQAL] I £0!" V‘G LA.) '5?ch , b) Find a basis for the column space of A. . 3 \ r \ O E L) e. LI. i , i i I j“ ’ Q5 \Urvm specs; c) Find a basis for the solution space of the homogeneous system AX z 0. 7““: 7(3‘21)? “\ Q” 0 h _ i , ’ n. * “ ' X5 0 4 l- ‘x \ j . 0 O O r. o ' 0 0 .. t r . 0 X ‘ I Xe}, X S (lbw. ‘ j . D j l Q'I'aé- UC/‘PI l3 q Lgsl‘) gov Sfisci (40 pts.) 2. Let S = {111, 212,113} be a basis for a vector space V and B = {U1,U2,U3}, C : {101,102,103} be subsets of V where ul ; 01 +112 +213, 1L2 2 111 + 122, U3 2 121 and w1=111+v2, w2=vl—v2, wgz—vl—vg+vg. a) Show that B is linearly independent. mi. and?” LL :6 UL met at»): c\\s.‘+LLub« cCt L23 : O , . gimca Kym“); are, “MEGA”; {a keg; L\-\'C.L\C_b';0 I “41:0 I (go ge L‘ t u) ck, : o I :1 a: Q , Waite, duo”) IQ; we/ (HAD \\NA€P€4<\,94L— b) Show that C spans V. Let v 6 S, (rice, (L 3> Q be: 5:; ’umya 0W"— "‘ \"9: L (7 UL sec-AA the \l : cuf‘tioxfr‘t é V3?- 0 (snslekr 4” “\WJ‘QM . fW '1‘ X1. “XI 33.: 5can “A (Add? \ C\\]\ TbULiCXJ : ¥\w‘ At¥1vut+¥2>wj> g L \I Q \J 7“ ~¥1 ‘Xb ‘ in W \I x was) '»v-\J.+\v‘, °* x .1 ._ L 1-1: 7- + ‘51, \_ ’L i " ‘-‘ [A X c) Find the change of coordinate matrix PBS, from B to 5'. WC VWVL 3 9 \ 3 \A’ v e kl)‘ W4“ k \ ' ‘ r O r. i; RB R‘QA‘ F; K t g ‘33 L3) “‘3 \ O i 6 ¢ (1) Find the change of coordinate matrix P55, from S to B. .. \ C9 0 a o \ r “$— \ 1 ‘ \ ° 6 “mfg L psi), ‘5 PE ; K L b O 1 0 41,7012. 0 i 0 ’0 \ ‘2) __ o \ "L 5% “ -\ -\ o “i e) Find 0 coordinate MC of a vector v E V if B = [ 1) Hi V A . . _ it w + ‘ \/ ‘: 0| 1—H + LED"? ‘CL $3 ,9 in“; ¥1/ V‘L /\L3 Sugb ‘H/lci' V ’ Xxwi‘f 7’ 1’ \wai . . i -\J -U,+\I 9_o\*\=>‘*c C1 Kikivtl*¥b\4&:wl+ug/\+Cvt : )L‘ Olf‘i'VL'X "i‘ XLLVi”\!’1)*¥$i ’ ‘1‘ R. /2'— l W V1 2 a], u+L+c ‘: mamas—x3 ‘M » a /2_ ' _ ~ I; . “*‘Q ~; XL "éu “‘13 \(‘b ' 3" CA [TA 1 ‘3: ‘ ‘~' (A i \ ' {L- \ 4 4 u o l 5 c, Jrkwn. \Jt an 642/ x“) Lokidd we ...
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MD term 2 - ., n.»— .WA '(30 pts.) 3. Let V be the...

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