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# final_s09_post - Math 235 - Final Exam Spring 2009 NOTE:...

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Math 235 - Final Exam Spring 2009 NOTE: The questions on this exam does not exactly reﬂect which questions will be on this terms exam. That is, some questions asked on this exam may not be asked on our exam and there may be some questions on our exam not asked here. 1. Short Answer Problems a) By considering the dimensions of the range or null space, determine the rank and the nullity of L : P 2 M (2 , 2) given by L ( ax 2 + bx + c ) = ± a a c c ² . b) Let V,W be ﬁnite dimensional vectors spaces over R . Give the formula for ﬁnding the matrix of a linear transformation L : V W with respect to any basis B for V and any basis C for W . c) Let A and B be n × n real matrices such that A = A T and B = - B T . Prove that A + iB is Hermitian. d) State the principal axis theorem. e) State Schur’s theorem. 2. Let ~ y = 5 - 9 5 and let W be the subspace of R 3 spanned by - 3 - 5 1 , - 3 2 1 . a) Find proj

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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 - Math 235 - Final Exam Spring 2009 NOTE:...

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