quiz2-solns - as T ( a 1 v 1 + + a n v n ) = 0 . Hence a 1...

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MATH 110 - SOLUTION TO QUIZ 2 LECTURE 1, SUMMER 2009 GSI: SANTIAGO CA ˜ NEZ Let V and W be vector spaces over a field F and suppose that T : V W is an injective linear map. Prove that if ( v 1 ,...,v n ) is linearly independent in V , then ( Tv 1 ,...,Tv n ) is linearly independent in W . Proof. Suppose that we have scalars a 1 ,...,a n F so that a 1 Tv 1 + ··· + a n Tv n = 0 . We must show that each a i is zero. Since T is linear, we can rewrite the above equation
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Unformatted text preview: as T ( a 1 v 1 + + a n v n ) = 0 . Hence a 1 v 1 + + a n v n null T . But T is injective, so null T = { } . Thus a 1 v 1 + + a n v n = 0 . Since v 1 ,...,v n are linearly independent, a i = 0 for each i as was to be shown. Date : July 2, 2009....
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