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 300: 13

Answer

Not linear.

Work Step by Step

In this case the transformation is linear if $T(v_1+v_2)=T(v_1)+T(v_2)$ and $T(av_1)=aT(v_1)$ for any $v_1,v_2$ in the domain and any $a$ scalar. Here this is violated because e.g. $T(1,1)=(1,1,1)$ but $T((1,1)+(1,1))=T(2,2)=(\sqrt2,4,\sqrt2)\ne(2,2,2)=(1,1,1)+(1,1,1)=T(1,1,+T(1,1).$
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