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assign7 - MATH 235 Inner Product Spaces Assignment 7 Hand...

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MATH 235: Inner Product Spaces, Assignment 7 Hand in questions 3,4,5,6,9,10 by 9:30 am on Wednesday March 26, 2008. Contents 1 Orthogonal Basis for Inner Product Space 2 2 Inner-Product Function Space 2 3 Weighted Inner Product in R 2 *** 2 4 Inner Product in S 3 *** 2 5 Inner Products in M 22 and P 2 *** 3 6 Cauchy-Schwarz and Triangle Inequalities *** 3 7 Length in Function Space 3 8 More on Function Space 3 9 Linear Transformations and Inner Products *** 4 10 MATLAB *** 5 1

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1 Orthogonal Basis for Inner Product Space If V = P 3 with the inner product < f, g > = integraltext 1 - 1 f ( x ) g ( x ) dx , apply the Gram-Schmidt algorithm to obtain an orthogonal basis from B = { 1 , x, x 2 , x 3 } . 2 Inner-Product Function Space Consider the vector space C [0 , 1] of all continuously differentiable functions defined on the closed interval [0 , 1]. The inner product in C [0 , 1] is defined by < f, g > = integraltext 1 0 f ( x ) g ( x ) dx. Find the inner products of the following pairs of functions and state whether they are orthogonal 1. f ( x ) = cos(2 πx ) and g ( x ) = sin(2 πx ) 2. f ( x ) = x and g ( x ) = e x 3. f ( x ) = x and g ( x ) = 3 x 3 Weighted Inner Product in R 2 *** Show whether or not the following are valid inner products in R 2 . If one is a valid inner product, then find a nonzero vector that is orthogonal to the vector y = ( 2 1 ) T . 1. ( u, v ) := 7 u 1 v 1 + 1 . 2 u 2 v 2 . 2. ( u, v ) := 7 u 1 v 1 1 . 2 u 2 v 2 . 3. ( u, v ) := 7 u 1 v 1 1 . 2 u 2 v 2 . 4 Inner Product in S 3 *** Let S 3 denote the vector space of 3 × 3 symmetric matrices. 1. Prove that ( S, T ) = tr ST is an inner product on S 3 . 2. Let S 1 = 1 0 1 0 1 0 1 0 0 and S 2 = 1 0 1 0 3 0 1 0 0 . Apply the Gram-Schmidt process in S 3 to find a matrix orthogonal to both S 1 and S 2 . 3. Find an orthonormal basis for S 3 using the above three matrices. 4. Let T = 1 0 1 0 1 1 1 1 1 . Find the projection of T on the span of { S 1 , S 2 } . 2
5 Inner Products in M 22 and P 2 *** Show which of the following are valid inner products. 1. Let A = bracketleftbigg a 1 a 2 a 3 a 4 bracketrightbigg and B = bracketleftbigg b 1 b 2 b 3 b 4 bracketrightbigg be real matrices. (a) Define ( A, B ) := a 1 b 1 + 2 a 2 b 3 + 3 a 3 b 2 + a 4 b 4 . (b) Define ( A, B ) := tr B T A. 2. Let p ( x ) and q ( x ) be polynomials in P 2 . (a) Define ( p, q ) := p ( 1) q ( 1) + p ( 1 2 ) q ( 1 2 ) + p ( 3) q ( 3). (b) Define ( p, q ) := p ( 3) q ( 3) + p ( 1 2 ) q ( 1 2 ) + p ( 1) q ( 1). 6 Cauchy-Schwarz and Triangle Inequalities *** 1. Let D = 3 0 0 0 π 0 0 0 2 . Consider the inner product defined on R 3 by: ( u, v ) = u T Dv . Let u = 2 3 7 and v = 4 8 9 . Verify that both inequalities hold.

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