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final_f09_post_ans - Math 235 Final F09 Answers 1 Short...

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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 ( λ ) = (3- λ )( λ 2- 4 λ + 5). Hence, the eigenvalues of A are λ = 3 and λ = 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 non-trivial, L is not one-to-one 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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final_f09_post_ans - Math 235 Final F09 Answers 1 Short...

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