Linear Algebra and Its Applications (5th Edition)

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

Chapter 1 - Linear Equations in Linear Algebra - 1.8 Exercises - Page 69: 2

Answer

$T(\vec{u})=\begin{bmatrix}0.5\\0\\-2\end{bmatrix}$ $T(\vec{v})=\begin{bmatrix}0.5a\\0.5b\\0.5c\end{bmatrix}$

Work Step by Step

$\begin{align}T(\vec{u})=\mathbf{A}\vec{u}&=\begin{bmatrix}0.5&0&0\\0&0.5&0\\0&0&0.5\end{bmatrix}\begin{bmatrix}1\\0\\-4\end{bmatrix}\\ &=0.5\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}1\\0\\-4\end{bmatrix}\\ &=0.5\begin{bmatrix}1\\0\\-4\end{bmatrix} \end{align}$ $\begin{align}T(\vec{v})=\mathbf{A}\vec{v}&=\begin{bmatrix}0.5&0&0\\0&0.5&0\\0&0&0.5\end{bmatrix}\begin{bmatrix}a\\b\\c\end{bmatrix}\\&=0.5\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}\begin{bmatrix}a\\b\\c\end{bmatrix}\\ &=0.5\begin{bmatrix}a\\b\\c\end{bmatrix} \end{align}$
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