3. Let TV → V be an operator on a finite-dimensional space V. (a) Prove that T is not invertible if and only if λ = 0 is an eigenvalue of T. (b) Suppose that T is invertible. Prove that if T is diagonalizable, then T−¹ is diagonalizable. ● If you divide by something in your proof, make sure to justify that you're not dividing by zero!

3. Let TV → V be an operator on a finite-dimensional space V. (a) Prove that T is not invertible if and only if λ = 0 is an eigenvalue of T. (b) Suppose that T is invertible. Prove that if T is diagonalizable, then T−¹ is diagonalizable. ● If you divide by something in your proof, make sure to justify that you're not dividing by zero!

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.2: Norms And Distance Functions

Problem 33EQ

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