Prove the matrix version of the corollary to Theorem 5.5: If A ∈Mn×n(F) has n distinct eigenvalues, then A is diagonalizable. Theorem 5.5. Let T be a linear operator on a vector space V, and let λ1, λ2, . . . , λk be distinct eigenvalues of T. If v1, v2, . . . , vk are eigenvectors of T such that λi corresponds to vi(1≤i ≤k), then {v1, v2, . . . , vk} is linearly independent.

Prove the matrix version of the corollary to Theorem 5.5: If A ∈Mn×n(F) has n distinct eigenvalues, then A is diagonalizable. Theorem 5.5. Let T be a linear operator on a vector space V, and let λ1, λ2, . . . , λk be distinct eigenvalues of T. If v1, v2, . . . , vk are eigenvectors of T such that λi corresponds to vi(1≤i ≤k), then {v1, v2, . . . , vk} is linearly independent.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter7: Distance And Approximation

Section7.1: Inner Product Spaces

Problem 11AEXP

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