Let V be a finite-dimensional vector space with ordered bases α, β, and γ.(a) Prove that if Q and R are the change of coordinate matrices that change α-coordinates into β-coordinates and β-coordinates into γ-coordinates, respectively, then RQ is the change of coordinate matrix that changes α-coordinates into γ-coordinates. (b) Prove that if Q changes α-coordinates into β-coordinates, then Q−1 changes β-coordinates into α-coordinates.

Let V be a finite-dimensional vector space with ordered bases α, β, and γ.(a) Prove that if Q and R are the change of coordinate matrices that change α-coordinates into β-coordinates and β-coordinates into γ-coordinates, respectively, then RQ is the change of coordinate matrix that changes α-coordinates into γ-coordinates. (b) Prove that if Q changes α-coordinates into β-coordinates, then Q−1 changes β-coordinates into α-coordinates.

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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