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(a) Find an orthonormal basis of the vector space P₂ of polynomials of degree at most 2 with respect to the inner product (f.,g) = [ f(x)g(x) dx

(a) Find an orthonormal basis of the vector space P₂ of polynomials of degree at most 2 with respect to the inner product (f.,g) = [ f(x)g(x) dx

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question 2 (a) Find an orthonormal basis of the vector space P₂ of polynomials of degree at most 2 with respect to the inner product (f. g) = [°¸ ƒ(x)g(x) dx Show your working. (b) Let A be the following matrix: A = 1 0 a 0 a 0 1 For a 0, find an eigendecomposition of A. (c) Consider the set W of all 2 x 2 matrices A such that both (1, 2) and (2,-1) are eigenvectors of A. Prove that W is a subspace of the space of all 2 x 2 matrices and find the dimension of W.
Transcribed Image Text:Question 2 (a) Find an orthonormal basis of the vector space P₂ of polynomials of degree at most 2 with respect to the inner product (f. g) = [°¸ ƒ(x)g(x) dx Show your working. (b) Let A be the following matrix: A = 1 0 a 0 a 0 1 For a 0, find an eigendecomposition of A. (c) Consider the set W of all 2 x 2 matrices A such that both (1, 2) and (2,-1) are eigenvectors of A. Prove that W is a subspace of the space of all 2 x 2 matrices and find the dimension of W.
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