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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 wi 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 [HI1 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 {ulyu2va‘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‘)
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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. &
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B LIP” Find a basis for the column space of B = 6 1 23 .
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C :31?” Suppose that the n x n Hiatrices C and D are similar. Prove that det(C) = det(D). .44; ea! hadENE 3$,\“Y”i“b~, {3 ENE”.
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2 M) Mg) m) —. dd (3) J a} Show that the set of 2 x 2 non—invertible matrices is not a vector space.
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M: 0 b  aLha]+[°£])=e([bﬂi :I “:3 A“. .J mus. m e 1;: m Cimvi U'ﬂbiéf mJinxlinT) NOT a vainr ffmﬁa. 1 3 2
E if” Let '= g g a . Suppose T(x) =Fx. Find abasis for Kai(T).
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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.
 Spring '11
 KeithEmmert

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