This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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....
View
Full Document
 Spring '08
 CELMIN
 Math, Matrices, Orthogonal matrix, Short Answer Problems, Hermitian, cn vn, real canonical form, cr vr

Click to edit the document details