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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.

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.

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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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 cER is an eigenvalue of T, then c=1 or c= -1. || 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.
Transcribed Image Text: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 cER is an eigenvalue of T, then c=1 or c= -1. || 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.
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