ef06k - MAS 3105 Final Quiz and Key June 22, 2006 Prof. S....

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June 22, 2006 Final Quiz and Key Prof. S. Hudson The unlabelled problems are 10 points each. 1) [15pts] Which of these subspaces ? (answer Yes or No to each one) { ( x 1 ,x 2 ,x 3 ) T | x 1 + x 2 = 1 } (in V = R 3 ) { ( x 1 ,x 2 ,x 3 ) T | x 1 = x 2 = x 3 } (in V = R 3 ) The set of all 2x2 lower triangular matrices (in V = R 2 x 2 ). The set of all 2x2 singular matrices (in V = R 2 x 2 ). The set of all polynomials in P 4 of degree 2. 2) Let S = span { (1 , 2 , 3 , 0) T , (1 , 0 , 0 , 1) T } , a subspace of R 4 . Find a basis of S . 3) [20pts] Use the following XDX - 1 factorization to quickly calculate these. You do not have to simplify very much in a) to d). Numbers like 5 4 are OK. You shouldn’t have to multiply matrices (but, for example, I would not accept “ A 2 A 5 ” as an answer to part b) A = 4 3 3 - 5 4 - 5 5 - 3 6 = 1 - 1 0 1 0 - 1 - 1 1 1 4 0 0 0 1 0 0 0 9 1 1 1 0 1 1 1 0 1 a) A - 1 b) A 7 c) e A d) det (A) e) Find a matrix
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ef06k - MAS 3105 Final Quiz and Key June 22, 2006 Prof. S....

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