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Unformatted text preview: Homework #10 Solutions p 348, #10 The keys to this exercise are the following. Lemma 1. Let V be a vector space over a field F . If { v 1 , . . . , v n } is a linearly independent set in V and w 6∈ h v 1 , . . . , v n i then { v 1 , . . . , v n , w } is linearly independent as well. Proof. Let a 1 , . . . , a n , b ∈ F so that a 1 v 1 + ··· + a n v n + bw = 0. If b 6 = 0 then we have w = ( b 1 a 1 ) v 1 + ··· + ( b 1 a n ) v n ∈ h v 1 , . . . , v n i which is a contradiction. It follows that b = 0 and so 0 = a 1 v 1 + ··· + a n v n + bw = a 1 v 1 + ··· + a n v n , which implies, via the linear independence of v 1 , . . . , v n , that a 1 = ··· = a n = 0. That is, the only linear combination of v 1 , . . . , v n , w that equals 0 is the trivial combination. Hence, { v 1 , . . . , v n , w } is linearly independent. Lemma 2. Let V be a vector space over a field F . If { v 1 , . . . , v n } is a basis for V and { w 1 , . . . , w m } is a linearly independent set in V then m ≤ n . Proof. The proof of Theorem 19.1 can be used, word for word. Now let S = { v 1 , v 2 , . . . , v n } be a set of linearly independent vectors in a finite dimensional vector space V . If h S i = V then S is a basis for...
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This note was uploaded on 02/29/2012 for the course MATH 4363 taught by Professor Ryandaileda during the Spring '07 term at Trinity University.
 Spring '07
 RyanDaileda
 Vector Space

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