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Unformatted text preview: Math 235 Final F09 Answers 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 } . b) Assume that ~v W and ~v W . By definition of W we have that h ~v,~s i = 0 for every ~s W . But, ~v W , so we have h ~v,~v i = 0 and hence ~v = ~ 0 since h , i is an inner product.is an inner product....
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This note was uploaded on 04/18/2010 for the course MATH 235 taught by Professor Celmin during the Spring '08 term at Waterloo.
 Spring '08
 CELMIN
 Math, Matrices

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