This preview shows pages 1–4. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 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 InnerProduct 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 CauchySchwarz 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 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 GramSchmidt algorithm to obtain an orthogonal basis from B = { 1 ,x,x 2 ,x 3 } . 2 InnerProduct 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 x ) and g ( x ) = sin(2 x ) 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 1 3 0 1 . Apply the GramSchmidt 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 1 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)....
View
Full
Document
This note was uploaded on 04/27/2010 for the course MATH 235 taught by Professor Celmin during the Winter '08 term at Waterloo.
 Winter '08
 CELMIN
 Math, Linear Algebra, Algebra

Click to edit the document details