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Test2Solutions - Math 332 w Test 2 Printed Name: C Please...

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Unformatted text preview: Math 332 w Test 2 Printed Name: C Please carefully work all of the following problem(5). You must SHOW YOUR WORK to receive ANY credit! Please answer the following questions. Let V be a vector space. Suppose W E V. Carefully define what it means "when we say W is a subspace of V. . [LU E: Co. Vie-{‘0’ 33‘3"“, NW7 Hi okWthW gm“ B in?” Let f : V w-i W be a mapping between any two vector spaces and suppose U Q W. Define, using set builder notation, j":l $qu] —_ {0 UN for) Uta] ~ Let V be a vector space and suppose B C V. Define what is meant when we say 5 is a basis. % .j a, (‘NMHH'IU ffl‘i‘ free} State the RmkuNullity Theorem. Fix—F nnj ‘Mmfii‘x; hth 0f cab..me €geu‘ {J #6 CLM—EnfaFI/‘n 0 6:18.. w‘mwmpact [HI-1 Hm omega“ #3,! H5; 4w.“ Iii—moi. is, An mm) A: 11% MWHL D3“ (Mum) E in?” Let f : V —l W be a. mapping between any two vector spaces. Define what it means when We say f is an isomorphism. 5: 11 shmawnPldh } u i~i f .‘1 own). Dri Emmert Spring 2010 A I of IV Points Earned :l of 35 Math 332 -— Test 2 Please carefuliy answer one of the following two sLatements Let V be a vecmr Space" Le" 3 g {ulyu2v---a‘1k} be any nonempty subset of V. Show that Span(S} is a. subspace of V. @- safim w; alarm), ,9 sham. “- i .a l‘ '9 a -/ “' f h‘) LEA “a IECSY ’32: “Ca; l?! fléuh fir HM. Hula-“J 9‘) J W'L'JJJU ‘flk . ", {uh . (F, F; r ‘3 _. C; “h _\N\b§u c“ G) a: *5] Warm“); : (di 19;)“; w) x+J éjr i 1 9.4! ,,." Cdme-b’xw‘ of «3er f’dw LCE‘AS be, Cwn‘j jLMkM-r. R - I; ; C1 [W‘hr' ® 59? Silva-«(33; J9“: =1§(Jo<;)31 3) Jr: Effie-“(‘0 JME 1x J I Cowhqugm D’f thkh—rl fifixfl". S\ I w) h a Jhlangg‘bt of) Let f I V —* W be a linear transformation between two vector Spaces and suppose U is a subspace of V. Show that f [U5 is a subspace of W. HEW—th 6 6% (“(0. Cu A 0~ JWLJJP‘W “f "j HSJEfLC‘Q/J” .4 ‘ 4- 9'3 - aauazeu' ‘3 {(lii3zw3 q, 3({U‘23'w2a. (a) may; 6 EEfl; cams; = {(3‘) gm) zfniggfifflfl. LL“: K l5!- Bv‘j JLALAF' (a) ch‘o‘ ewe; as} was «5341) e; Eu]. E‘M H0va U a JuLJfALi’, df Points Earned l: of 15 r ' II of IV Dr. Emmett Spring 2010 Math 332 — Test 2 Please carefully answer one of the following two statements Suppose T :_ —> W is a. linear transformation between two vector Spaces. Suppose V has hafiis B = {131,132,133}, W has basis C = {c1422}, and T(b1) a gal — 3C21T(b2)= 531 + 732’ T(b3) = 4c1 — 6C2. Find the matrix M such that ET(x)]C = M[x]8. m 3 [a “£913{Lwcslgimfiflgj .. zsbr "[‘37—6 Suppose L is 8- lifle through the Origin Of R". Show that L is a one dimensional subSpace of IR“. . F} P) a t i (X? t (x tfpan({\71)qomii LL44. a. Ltfiww y/ a, 1 W] U ,Mtfbtmi ane Mot Dr. Emmert Spring 2010 III of TV Points Earned E of 15 Math 332 — Test 2 Complete the following short—answer questions. A Lip” Let x 7» (—15,35,2)T and suppose we use the basis 8 = {(1,2,3)T,(—2,4,—1)T,(2,—~2,0)T}. Find [x]5. & - Z .. {W} l b 0 3 [2, ‘ivz 3‘; “’4 O ' ‘9 6 12 2 46 B LIP” Find a basis for the column space of B = 6 1 23 . 3 1 14 12 1% mi ‘03 ‘7- 1 . L; -w‘ _e\;3-'""70‘5 ‘35 931 ‘JHQWMJ‘L 3 7 l H o o O 3 1 C :31?” Suppose that the n x n Hiatrices C and D are similar. Prove that det(C) = det(D). .44; ea! had-ENE 3$,\“Y”i“-b~, {3 ENE”. J? M83 aid (36?“ we) MB) we“) ' 2 M) Mg) m) —. dd (3) J a} Show that the set of 2 x 2 non—invertible matrices is not a vector space. 0 o H F i O l O :- Jt£ 0‘ flwfub) M: 0 b - aLha]+[°£])=e([bfli :I “:3 A“. .J mus. m e 1;: m Cimvi U'flbiéf mJinxlinT) NOT a vain-r ffmfia. 1 3 2 E if” Let '= g g a . Suppose T(x) =Fx. Find abasis for Kai-(T). 4:, rd .Q-a b 121 foEKuLTLif/H , Fifi L“) 01 l 0 Ix z’X ' . w ; . , ,E 0 o "*3 I 3 a _ “<3 # i 5 L511 (J $533 5! xlr‘xj 'X'" "9‘3 :363 '3 '7 a) i . (X :7”? “X3 Points Earned I: of 35 IV of IV Dr. Emmert Spring 2010 ...
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This note was uploaded on 01/16/2012 for the course MATH 332 taught by Professor Keithemmert during the Spring '11 term at Tarleton.

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Test2Solutions - Math 332 w Test 2 Printed Name: C Please...

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