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Suppose V is finite-dimensional, T E L(V) has n = dim V distinct eigenvalues, and S E L(V) has the same eigenvectors as T (not necessarily with the same eigenvalues). Prove that ST = TS. (Hint: find an eigenbasis for %3| V.)

Suppose V is finite-dimensional, T E L(V) has n = dim V distinct eigenvalues, and S E L(V) has the same eigenvectors as T (not necessarily with the same eigenvalues). Prove that ST = TS. (Hint: find an eigenbasis for %3| V.)

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.1: Eigenvalues And Eigenvectors
Problem 7E
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Suppose V is finite-dimensional, T E L(V) has n = dim V distinct eigenvalues, and SE L(V) has the same eigenvectors as T (not necessarily with the same eigenvalues). Prove that ST = TS. (Hint: find an eigenbasis for V.)
Transcribed Image Text:Suppose V is finite-dimensional, T E L(V) has n = dim V distinct eigenvalues, and SE L(V) has the same eigenvectors as T (not necessarily with the same eigenvalues). Prove that ST = TS. (Hint: find an eigenbasis for V.)
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