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Unformatted text preview: Selected answers to assignment 7: 2.5, 3.1 2.5 9. “is similar to” is an equivalence relation: reflexive: A = I 1 AI symmetric: if B = Q 1 AQ , then A = ( Q 1 ) 1 B ( Q 1 ). transitive: if B = Q 1 AQ and C = P 1 BP , then C = P 1 Q 1 AQP = ( QP ) 1 A ( QP ). 12. The part of the corollary requiring proof is that the equality holds. Q is the change of coordinate matrix from γ coordinates to β (= stan dard basis) coordinates automatically (see the top of page 112). The equality’s proof comes from remembering L A is defined specifically with respect to the standard basis β and applying Theorem 2.23. 13. If β is a basis, then definitionally Q is the changeofcoordinate matrix. Hence we need only prove β is a basis. Suppose there is a nontrivial representation of zero from elements of β ; in other words, a 1 x 1 + a 2 x 2 + . . . + a n x n = for some a i not all zero. Then by definition of β , we get a 1 n X i =1 Q i 1 x i + a 2 n X i =1 Q i 2 x i + . . . + a n n X i =1...
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 Winter '07
 Liu
 Linear Algebra, Algebra, Standard basis, Q−1 AQ, ith row, aj Qij, aj Qnj

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