16 (16) Let T be a linear operator on a vector space V over the field F, and let g(t) bea polynomial over F. Prove that if x is an eigenvector of T with correspondingeigenvalue A, then g(T)(x) = g(A)(x). That is, x is an eigenvector of g(T) withcorresponding eigenvalue g(A).

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Answered: (16) Let T be a linear operator on a…

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 8E

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(16) Let T be a linear operator on a vector space V over the field F, and let g(t) be a polynomial over F. Prove that if x is an eigenvector of T with corresponding eigenvalue A, then g(T)(x) corresponding eigenvalue g(A). g(A)(x). That is, x is an eigenvector of g(T) with