assig4 - MATH 245 Linear Algebra 2, Assignment 4 Due Wed...

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Unformatted text preview: MATH 245 Linear Algebra 2, Assignment 4 Due Wed Nov 10 1: (a) Let A 1 = 1 2- 1 0 , A 2 = 3 4- 1 2 and A 3 = 3- 1- 2 4 . Apply the Gram-Schmidt Procedure to the basis U = A 1 ,A 2 ,A 3 to obtain an orthonormal basis for U = Span A 1 ,A 2 ,A 3 M 2 2 . (b) Find an orthonormal basis for P 2 with the inner product given by p,q = p (0) q (0)+ p (1) q (1)+ p (2) q (2) by applying the Gram-Schmidt Procedure to the standard basis 1 ,x,x 2 . 2: Consider P 2 as a subspace of C [- 1 , 1] with its standard inner product, which is given by f,g = Z 1- 1 fg . Applying the Gram-Schmidt Procedure to the standard basis 1 ,x,x 2 for P 2 yields the orthogonal basis p ,p 1 ,p 2 , where p = 1, p 1 = x and p 2 = x 2- 1 3 . (a) Find the polynomial f P 2 which minimizes Z 1- 1 ( f ( x )- | x | ) 2 dx . (b) Let f C [- 1 , 1]. Given that 1 = Z 1- 1 f ( x ) dx = Z 1- 1 xf ( x ) dx = Z 1- 1 x 2 f ( x ) dx , find the minimum possible value for Z 1- 1 f ( x ) 2 dx ....
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This note was uploaded on 12/08/2011 for the course MATH 245 taught by Professor New during the Fall '09 term at Waterloo.

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