final_s09_post_ans

final_s09_post_ans - Math 235 Final S09 Answers 1. a) A...

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Unformatted text preview: Math 235 Final S09 Answers 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) If A~v = λ~v , with ~v 6 = ~ 0 then ~v is an eigenvector of A with eigenvalue λ . 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 ,- 5 , 1) + (6 ,- 4 ,- 2) = (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 >- = <- proj W ~v,- proj W ~v > + < ~v,~v > +( ~v,- proj W ~v > = < ~v,~v >- < ~v- proj W ~v,- proj W ~v > = k ~v k...
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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.

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final_s09_post_ans - Math 235 Final S09 Answers 1. a) A...

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