ps8 - Problem Set 8 Math 115A/5 Fall 2011 Due Friday...

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Problem Set 8 Math 115A/5 — Fall 2011 Due: Friday, December 2 Problem 1 (6.2.2) In each part, apply the Gram-Schmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for span( S ). Then normalize the vectors in the basis to obtain an orthonormal basis β for span( S ), and compute the Fourier coefficients of the given vector relative to β . (c) V = P 2 ( R ), with inner product h f ( x ) , g ( x ) i = R 1 0 f ( t ) g ( t ) dt , S = { 1 , x, x 2 } , and h ( x ) = 1 + x . (d) V = span( S ), where S = { (1 , i, 0) , (1 - i, 2 , 4 i ) } , and x = (3 + i, 4 i, - 4). (g) V = M 2 × 2 ( R ), S = 3 5 - 1 1 , - 1 9 5 - 1 , 7 - 17 2 - 6 , and A = - 1 27 - 4 8 . Problem 2 (6.2.4) Let S = { (1 , 0 , i ) , (1 , 2 , 1) } in C 3 . Compute S . Problem 3 (6.2.6) Let V be an inner product space, and let W be a finite-dimensional subspace of V . If x / W , prove that there exists y V such that y W , but h x, y i 6 = 0. Problem 4 (6.2.8) Prove that if { w 1 , w 2 , . . . , w n } is an orthogonal set of nonzero vectors then the vectors v 1 , v 2 , . . . , v n derived from the Gram-Schmidt process satisfy v i = w i for i = 1 , 2 , . . . , n . Problem 5 (6.2.13) Let V be an inner product space, S and S 0 be subsets of V , and W be a finite-dimensional subspace of V . Prove the following results: (a) S 0 S implies that S S 0 . (b) S ( S ) ; so span( S ) ( S ) . (c) W = ( W
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