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# solns7 - MATH 235 Inner Product Spaces SOLUTIONS to Assign...

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Unformatted text preview: MATH 235: Inner Product Spaces, SOLUTIONS to Assign. 7 Questions handed in: 3,4,5,6,9,10. Contents 1 Orthogonal Basis for Inner Product Space 2 2 Inner-Product Function Space 2 3 Weighted Inner Product in R 2 *** 2 3.1 SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 4 Inner Product in S 3 *** 3 4.1 SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 5 Inner Products in M 22 and P 2 *** 4 5.1 SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 Cauchy-Schwarz and Triangle Inequalities *** 5 6.1 SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 7 Length in Function Space 6 8 More on Function Space 7 9 Linear Transformations and Inner Products *** 8 9.1 SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 10 MATLAB *** 9 10.1 SOLUTIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 10.1.1 MATLAB Program for Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 12 10.1.2 Output of MATLAB Program for Solution . . . . . . . . . . . . . . . . . . . 13 1 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 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 . 3.1 SOLUTIONS BEGIN SOLUTION: Note that in each case, the inner product can be written as ( u,v ) = u T Dv , for an appropriate diagonal matrix D . We see that ( u,v ) = u T Dv = ( u T Dv ) T = v T Du = ( v,u ) . And, ( αu,v + w ) = α ( u T D ( v + w )) = α ( ( u,v ) + ( u,w ) ). Therefore, the first three properties for an inner product all hold true. 1. For ( u,v ) := 7 u 1 v 1 + 1 . 2 u 2 v 2 , the diagonal matrix D = parenleftbigg 7 0 1 . 2 parenrightbigg . Therefore, ( u,u ) := 7 u 2 1 +1 . 2 u 2 2 ≥ 0, with equality if and only if the vector u = 0, i.e. this is a valid innerproduct....
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solns7 - MATH 235 Inner Product Spaces SOLUTIONS to Assign...

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