Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

5th Edition

ISBN: 9780134689531

Author: Lee Johnson, Dean Riess, Jimmy Arnold

Publisher: PEARSON

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Textbook Question

Chapter 5.6, Problem 29E

A sequence of orthogonal polynomials usually satisfies a three-term recurrence relation. For example, the Chebyshev polynomials are related by

$T n + 1 ( x ) = 2 x T n ( x ) − T n − 1 ( x )$ , $n = 1 , 2 , ...$ ,

where $T 0 ( x ) = 1$ and $T 1 ( x ) = x$ . Verify that the polynomials defined by the relation (R) above are indeed orthogonal in $C [ − 1 , 1 ]$ with respect to the inner product in Exercise 28. Verify this as follows:

a) Make the change of variables $x = cos θ$ , and use induction to show that $T k ( cos θ ) = cos k θ$ , $k = 0 , 1 , ....$ where $T k ( x )$ is defined by (R).

b) Using part a), show that $〈 T i , T j 〉 = 0$ when $i ≠ j$ .

c) Use induction to show that $T k ( x )$ is a polynomial of degree $k$ , $k = 0 , 1 , ...$ .

d) Use (R) to calculate $T 2 , T 3 , T 4 ,$ and $T 5$ .

28. An inner product on $C [ − 1 , 1 ]$ is

$〈 f , g 〉 = 2 π ∫ − 1 1 f ( x ) g ( x ) 1 − x 2 d x$ .

Starting with the set ${ 1 , x , x 2 , x 3 , ... }$ , use the Gram-Schmidt process to find polynomials $T 0 ( x ) , T 1 ( x ) , T 2 ( x )$ , and $T 3 ( x )$ such that $〈 T i , T j 〉 = 0$ when $i ≠ j$ . These polynomials are called the Chebyshev polynomials of the first kind. [Hint: Make a change of variables $x = cos θ$ .]

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Chapter 5.6, Problem 28E

Chapter 5.6, Problem 30E

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Chapter 5 Solutions

Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - For u,v and w given in given Exercises 13,...Ch. 5.2 - Prob. 4E Ch. 5.2 - Prob. 5E Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - Prob. 7E Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...Ch. 5.2 - In Exercises 6-11, the given set is a subset of a...

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Solution Summary: The author explains that the polynomials defined by the relation (R) are orthogonal in Cleft[-1,1right] with respect to the inner product given and the proof that T

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