Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 6 - Linear Transformations - 6.3 The Kernel and Range of a Linear Transformation - Problems - Page 407: 23

Answer

See below

Work Step by Step

For every $v \in V$ there exist scalars $a_1,a_2,...,a_k$ such as: $v=a_1v_1+a_2v_2+...+a_kv_k$ We obtain: $T(v)=T(a_1v_1+a_2v_2+...+a_kv_k)\\ =a_1T(v_1)+a_2T(v_2)+...+a_kT(v_k)\\ =a_1S(v_1)+a_2S(v_2)+...+a_kS(v_k)\\ =S(a_1v_1+a_2v_2+...+a_kv_k)\\ =S(v)$ with $v \in V$ Hence, $T=S$
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