22. Let T be a linear operator on a finite-dimensional vector space V over the field F. Prove that if g(t) e P(F) and x is an eigenvector of T corresponding to eigenvalue 2, then g(TXx) = g(2)x. 23. Use Exercise 22 to prove that if f(t) is the characteristic polynomial of a diagonalizable linear operator T, then f(T) = To, the zero operator.

22. Let T be a linear operator on a finite-dimensional vector space V over the field F. Prove that if g(t) e P(F) and x is an eigenvector of T corresponding to eigenvalue 2, then g(TXx) = g(2)x. 23. Use Exercise 22 to prove that if f(t) is the characteristic polynomial of a diagonalizable linear operator T, then f(T) = To, the zero operator.

Name: Please see image 22. Let T be a linear operator on a finite-dimensiona
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Description: Please see image 22. Let T be a linear operator on a finite-dimensional vector space V over thefield F. Prove that if g(t) e P(F) and x is an eigenvector of T correspondingto eigenvalue 2, then g(TXx) = g(2)x.23. Use Exercise 22 to prove that if f(t)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.3: Eigenvalues And Eigenvectors Of N X N Matrices

Problem 28EQ

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