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assig8M235W08 - Math 235 Linear Algebra II Assignment 8 Due...

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Math 235 Linear Algebra II Assignment 8 Due 9:30am, Wednesday Nov 12, 2008, in the drop box designated for your section. 1. Let W be the subspace of R 4 spanned by (1 , 0 , - 1 , 2) and (2 , 1 , 0 , - 1) and take the inner product to be the dot product. (a) Find the dimension of W . (b) Find the vector in W which is closest to the vector (4 , 3 , 2 , 1). (c) Find a basis for W . (d) Find the unique w 1 W and w 2 W such that (4 , 3 , 2 , 1) = w 1 + w 2 . 2. Consider the real vector space C [0 , 1] of continuous functions on inter- val [0 , 1] together with the inner product given by < f, g > = R 1 0 f ( t ) g ( t ) dt for f, g C [0 , 1]. Let P 2 ( R ) be subspace of C [0 , 1] having a basis given by B = { 1 , x, x 2 } . Find the best quadratic approximation to x on [0 , 1], i.e. find the best approximation to x by elements of P 2 ( R ). 3. Find the least-squares approximation to a solution of Ax = b by con- structing the normal equations for ˆ x and then solving for ˆ x for the following cases. Also find the least-squares error associated with each approximation.
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