final_s09_post_ans

# Final_s09_post_ans - Math 235 Final S09 Answers NOTE These are only answers to the problems and not full solutions On the nal exam you will be

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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 ² 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,

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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.

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Final_s09_post_ans - Math 235 Final S09 Answers NOTE These are only answers to the problems and not full solutions On the nal exam you will be

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