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Unformatted text preview: Math 235 Final F09 Answers NOTE: These are only answers to the problems and not full solutions! On the final exam you will be expected to show all steps used to obtain your answer. 1. Short Answer Problems a) A * = 3 i i 2 1 . b) A is Hermitian since A * = A , and hence A is normal. c) Let Q and P be orthogonal matrices so that Q 1 = Q T and P 1 = P T . Then, ( PQ ) 1 = Q 1 P 1 = Q T P T = ( PQ ) T . Hence, PQ is orthogonal. d) Observe that the characteristic polynomial of A is C ( ) = (3 )( 2 4 + 5). Hence, the eigenvalues of A are = 3 and = 2 i . Thus, A is already in real canonical form with P = I . 2. a) nullity L = 1 and rank L = 2. b)Since null( L ) is nontrivial, L is not onetoone and hence it is not an isomorphism. On the other hand, since dim( R 3 ) = dim( W ) = 3, we have that R 3 and W are isomorphic. c) 1 1 1 0 1 0 1 1 3. a) W = { ~x V  h ~x, ~w i = 0 for every ~w W } ....
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Math

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