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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.

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter5: Inner Product Spaces
Section5.3: Orthonormal Bases:gram-schmidt Process
Problem 17E: Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner...
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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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