Linear Algebra and Its Applications (5th Edition)

Published by Pearson
ISBN 10: 032198238X
ISBN 13: 978-0-32198-238-4

Chapter 1 - Linear Equations in Linear Algebra - 1.9 Exercises - Page 80: 36

Answer

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Work Step by Step

Concepts Definition of Linear Transformation Algebraic Properties of $\mathbb{R}^{n}$ Plan Compute $T(S(c \mathbf{u}+d \mathbf{v}))$ for $\mathbf{u}, \mathbf{v}$ and scalars $c$ and $d$ Solve Take $\mathbf{u}$ and $\mathbf{v}$ in $\mathbb{R}^{p}$ and let $c$ and $d$ be scalars. Then: $T(S(c \mathbf{u}+d \mathbf{v}))=T(c \cdot S(\mathbf{u})+d \cdot S(\mathbf{v}))$ because $S$ is linear $=c \cdot T(S((u))+d \cdot T(S(\mathbf{v})) \text { since } T \text { is linear } r$ Conclusion The mapping $\mathbf{x} \rightarrow T(S(\mathbf{x}))$ is linear.
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