A5_soln - Math 235 Assignment 5 Solutions 1. On M (2 , 2)...

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Unformatted text preview: Math 235 Assignment 5 Solutions 1. On M (2 , 2) define the inner product < A,B > = tr( B T A ) and let S = Span 1 0 0 1 , 0 1 1 0 , 1- 1 1 . a) a) Use the Gram-Schmidt procedure to produce an orthonormal basis for S . Solution: We let ~v 1 = 1 0 0 1 and ~v 2 = 0 1 1 0 . We observe that h ~v 1 ,~v 2 i = 0, hence they are already orthogonal. Then ~v 3 = 1- 1 1- 2 2 1 0 0 1-- 1 2 0 1 1 0 =- 1 / 2 1 / 2 . Normalize each matrix and we get an orthonormal basis 1 2 1 0 0 1 , 1 2 0 1 1 0 , 1 2- 1 1 . b) Consider A = 2- 1 2 . Find the matrix B in S such that k A- B k is minimized. Solution: This matrix B is the projection of A onto S . Using the orthonormal basis from part (a), we have B = 2 2 1 2 1 0 0 1 + 1 2 1 2 0 1 1 0 +- 3 2 1 2- 1 1 = 1 0 0 1 + 1 / 2 1 / 2 + 3 / 2- 3 / 2 = 1 2- 1 1 . 2. Determine the orthogonal compliment of the subspace S = Span { x 2 + 1 } of P 2 under the inner product h p,q i = p (- 1) q (- 1) + p (0) q (0) + p (1) q (1)....
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A5_soln - Math 235 Assignment 5 Solutions 1. On M (2 , 2)...

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