Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 6 - Linear Transformations - 6.1 Introduction to Linear Transformations - 6.1 Exercises - Page 301: 52

Answer

$T$ is a linear transformation.

Work Step by Step

$T$ is a linear transformation. Let $A,B$ be two elements of the domain $M_{n,n}$ and $a$ be any arbitrary scalar. Then $T(aA+B)=(aA+B)X-X(aA+B)\\ =aAX+BX-XaA-XB\\ =aAX-XaA+BX-XB\\ =a(AX-XA)+BX-XB\\ =aT(A)+T(B)$
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