M235A7 - MATH 235 Inner Product Spaces Assignment 7 Hand in...

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MATH 235 Inner Product Spaces Assignment 7 Hand in questions 4,5,6,7,8,9 by 9:30 am on Wednesday November 15, 2006. 1. If V = P 3 with the inner product < f,g > = R 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. 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 > = R 1 0 f ( x ) g ( x ) dx. Find the inner products of the following pairs of functions and state whether they are orthogonal (a) f ( x ) = cos(2 πx ) and g ( x ) = sin(2 πx ) (b) f ( x ) = x and g ( x ) = e x (c) f ( x ) = x and g ( x ) = 3 x 3. Show that < u , v > = u 1 v 1 + 8 u 2 v 2 defines an inner product in R 2 . Under this inner product, find a vector y which is orthogonal to x = (1 , 1) T . 4. Define an inner product on R 2 by < ( x,y ) , ( x 0 ,y 0 ) > = 2 xx 0 - ( xy 0 + yx 0 ) + 2 yy 0 and let e 1 = (1 , 0) and e 2 = (0 , 1). (a) Prove directly that < v , v > 0 for all v R 2 . (b) Find || e 1 || , || e 2 || and < e 1 , e 2 > with respect to the inner product defined above. (c) Apply the Gram-Schmidt process to e 1 , e 2 to obtain a basis for R 2 which is orthonormal with respect to the inner product defined above. 5. (a) Let A = ± a 1 a 2 a 3 a 4 ² and B = ± b 1 b 2 b 3 b 4 ² be real matrices. Show that < A,B > = a 1 b 1 + a 2 b 3 + a 3 b 2 + a 4 b 4 is not an inner product on M 22 . (b) Let p ( x ) and q ( x ) be polynomials in P 2 . Show that < p,q > = p (0) q (0) + p ( 1 2 ) q ( 1 2 ) + p (1) q (1) is an inner product on P 2 . 6. Let W = Span { 1 , cos( x ) , cos(2 x ) } ∈ C [ - π,π ] with inner product < f,g > = R π - π f ( x ) g ( x ) dx . It is known that { 1 2 π , 1 π cos( x ) , 1 π cos(2 x ) } is an or- thonormal basis for W . A function f W satisfies R π - π f ( x ) dx = a , R π - π f ( x ) cos( x ) dx = b and R π - π f ( x ) cos(2 x ) dx = c . Find an expression for || f || in terms of a,b and c . 1
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7. Consider the vector space P 2 consisting of polynomials of degree at most 2 together with the inner product < f,g > = Z 1 0 f ( x ) g ( x ) dx , f,g P 2 . Let
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M235A7 - MATH 235 Inner Product Spaces Assignment 7 Hand in...

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