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) = [ f(x)g(x) dx Show your working. (b) Let A be the following matrix: A 10 a 010 a 01 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.
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) = [ f(x)g(x) dx Show your working. (b) Let A be the following matrix: A 10 a 010 a 01 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.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices
Problem 29EQ
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