Linear algebra 5. (a) Let V be an inner product space with F = R, and let T : V → V be an isomorphism of innerproduct spaces. Show that if ceR is an eigenvalue of T, then c= 1 or c = -1.(b) Suppose V is a finite-dimensional inner product space with Fa linear transformation such that V has an orthonormal basis {v1, v2, . .. , Vn} of eigenvectors ofT. Furthermore, assume that all eigenvalues of T come from the set {-1,1}. Prove that T is anisomorphism of inner product spaces.R, and suppose T : V → V is

For the first question we will use the key property of isomorphism that the linear transform if the…

(a) Let V be an inner product space with F = R, and let T : V → V be an isomorphism of inner product spaces. Show that if c E R is an eigenvalue of T, then c= 1 or c = -1. (b) Suppose V is a finite-dimensional inner product space with F = R, and suppose T : V → V is a linear transformation such that V has an orthonormal basis {v1, v2, ..., vn} of eigenvectors of T. Furthermore, assume that all eigenvalues of T come from the set {-1,1}. Prove that T is an isomorphism of inner product spaces.

(a) Let V be an inner product space with F = R, and let T : V → V be an isomorphism of inner product spaces. Show that if c E R is an eigenvalue of T, then c= 1 or c = -1. (b) Suppose V is a finite-dimensional inner product space with F = R, and suppose T : V → V is a linear transformation such that V has an orthonormal basis {v1, v2, ..., vn} of eigenvectors of T. Furthermore, assume that all eigenvalues of T come from the set {-1,1}. Prove that T is an isomorphism of inner product spaces.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 27EQ

See similar textbooks