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MA353_HW3_sol

MA353_HW3_sol - Purdue University MA 353 Linear Algebra II...

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Purdue University MA 353: Linear Algebra II with Applications Homework 3, due Feb. 5, solutions (I) Sec. 1.4 #13: Show that if S 1 and S 2 are subsets of a vector space V such that S 1 S 2 , then span S 1 span S 2 . In particular, if S 1 S 2 and span S 1 = V , deduce that span S 2 = V . Proof. Assume S 1 S 2 . Let x span S 1 . Then x = a 1 v 1 + a 2 v 2 + · · · + a n v n for some v i S 1 and a i F . But since S 1 S 2 , we know that v i S 2 . Hence x span S 2 . In particular, if span S 1 = V , then V span S 2 , which implies span S 2 = V . Sec. 1.5 #1 (b): Any set containing the zero vector is linearly dependent. Answer : Yes. Proof. Let S = { 0 V , v 1 , v 2 , . . . , v n } be a set containing the zero vector. Then consider 1 · 0 V + 0 F v 1 + 0 F v 2 + · · · + 0 F v n = 0 V . The left hand side is a nontrivial linear combination of the vectors in S , which equals the zero vector. Hence S is linearly dependent. #9: Let u and v be distinct vectors in a vector space V . Show that { u, v } is linearly dependent if and only if u or v is a multiple of the other.
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