final_s09_post_ans

final_s09_post_ans - Math 235 Final S09 Answers NOTE These...

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Math 235 Final S09 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. a) A basis for the nullspace is { x } , hence the nullity of L is 1. Thus, by the dimension theorem we have rank L + 1 = dim P 2 = 3 , hence rank L = 2. b) Let B = { ~v 1 , . . . ,~v n } then [ L ] = [ L ( ~v 1 )] C · · · [ L ( ~v n )] C c) ( A + iB ) * = A * - iB * = A T - iB T = A + iB , since A and B are real. Hence A is A + iB is Hermitian. d) A is symmetric if and only if A is orthogonally diagonalizable. e) State Schur’s theorem. Every square matrix is unitarily similar to a upper triangular matrix T where the diagonal entries of T are the eigenvalues of A . 2. a) proj W ~ y = 3 - 9 - 1 b) 40 . 3. k proj W ~v k 2 + k ~v - proj W ~v k 2 = < proj W ~v, proj w ~v > + < ~v - proj w ~v,~v - proj W ~v > = < proj W ~v, proj W ~v > + < ~v,~v - proj W ~v > - < proj W ~v,~v - proj W ~v > = < proj W ~v, proj W ~v > + < ~v,~v - proj W ~v > - 0 = < - proj W ~v, - proj W ~v > + < ~v,~v > +( ~v, - proj W ~v > = < ~v,~v > - < ~v - proj W ~v, - proj W ~v > = k ~v k 2 - 0 4. a) Q ( x 1 , y 1 ) = 3 x 2 1 + 16 y 2 1 , Q is positive definite, P = - 3 / 13 2 / 13 2 / 13
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