Suppose that L1 : V → W and L2 : W → Z are linear transformations and E, F, and G are ordered bases for V, W, and Z, respectively. Show that, if A represents L1 relative to E and F and B represents L2 relative to F and G, then the matrix C = BA represents L2 ◦ L1: V → Z relative to E and G. Hint: Show that BA[v]E = [(L2 ◦ L1)(v)]G for all v ∈ V.

Suppose that L1 : V → W and L2 : W → Z are linear transformations and E, F, and G are ordered bases for V, W, and Z, respectively. Show that, if A represents L1 relative to E and F and B represents L2 relative to F and G, then the matrix C = BA represents L2 ◦ L1: V → Z relative to E and G. Hint: Show that BA[v]E = [(L2 ◦ L1)(v)]G for all v ∈ V.

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 18EQ

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