2.5+3.1

2.5+3.1 - Selected answers to assignment 7 2.5 3.1 2.5 9...

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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 change-of-coordinate 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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2.5+3.1 - Selected answers to assignment 7 2.5 3.1 2.5 9...

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