Let V be a real vector space, and let T : V → V be a linear transfor- mation. Suppose that v₁ and v2 are eigenvalues of T corresponding to distinct eigenvalues A₁ A₂. (b) Now suppose additionally that V is an inner product space, and that T is a symmetric linear transformation: (Tv, w) = (v, Tw) for all v, w E V. Show that the eigenvectors v₁ and v2 are or- thogonal.

Let V be a real vector space, and let T : V → V be a linear transfor- mation. Suppose that v₁ and v2 are eigenvalues of T corresponding to distinct eigenvalues A₁ A₂. (b) Now suppose additionally that V is an inner product space, and that T is a symmetric linear transformation: (Tv, w) = (v, Tw) for all v, w E V. Show that the eigenvectors v₁ and v2 are or- thogonal.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.6: The Matrix Of A Linear Transformation

Problem 43EQ

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