Let V be a finite-dimensional vector space with dim V =n > 1, and let To : V → V denote the linear transformation defined by To(v) = 0 for all v E V. Prove that if T: V → V is any linear transformation with T² = To, then T is not invertible.

Let V be a finite-dimensional vector space with dim V =n > 1, and let To : V → V denote the linear transformation defined by To(v) = 0 for all v E V. Prove that if T: V → V is any linear transformation with T² = To, then T is not invertible.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section: Chapter Questions

Problem 16RQ

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Question

Let V be a finite-dimensional vector space with dim V =n > 1, and let To : V → V denote the linear
transformation defined by To(v) = 0 for all v E V. Prove that if T: V → V is any linear transformation with
T² = To, then T is not invertible.

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