2.42.5 - 2.4, Problem 7 Let A be an n n matrix. Theorem. If...

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§ 2.4, Problem 7 Let A be an n × n matrix. Theorem. If A 2 =0 ,then A is not invertible Proof. For a contradiction assume that A 2 =0and A is invertible. Then A has an inverse A - 1 and A 2 , A - 1 AA = A - 1 0 , IA , A . But then A - 1 A = A - 1 0=0 6 = I – Contradiction. Theorem. If AB for some nonzero n × n matrix B A is not invertible. Proof. Again, by contradiction. Assume AB A is invertible. Then AB , A - 1 AB = A - 1 0 , IB , B . Contradiction: we assumed B was nonzero. § 2.4, Problem 15 Theorem. Let V and W be Fnite-dimensional vector spaces, and let T : V W be a linear transformation. Suppose that β = { β 1 ,...,β n } is a basis for V .Then T is an isomorphism if and only if T ( β ) is a basis for W . Proof. ( ) First assume that T is an isomorphism. Since T an isomorphism it is one-to-one and onto. Problem 14(c) on page 75 applies to tell us that T ( β ) is a basis. ( ) Conversely, assume that T ( β )isabas isfor W . (We want to prove that T is an isomorphism. Since we already know T is linear we need prove that T is one-to-one and onto.) First we’ll prove T is one-to-one. Let
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This note was uploaded on 11/30/2009 for the course MATH 115A taught by Professor Liu during the Winter '07 term at UCLA.

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2.42.5 - 2.4, Problem 7 Let A be an n n matrix. Theorem. If...

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