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Found in: Page 54

### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

# Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation ${\mathbit{T}}\stackrel{\mathbf{\to }}{\left(x\right)}{\mathbf{=}}{\mathbit{A}}\stackrel{\mathbf{\to }}{\mathbf{x}}$ on this face.26. $\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

By an angle of ${90}^{\circ }$ , the face gets rotated clockwise.

See the step by step solution

## Step by Step Explanation: Step 1: Consider the matrix

Let the matrix be,

$A=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$

Then,

$T\stackrel{\to }{\left(x\right)}=A\stackrel{\to }{x}\phantom{\rule{0ex}{0ex}}T\stackrel{\to }{\left(x\right)}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\stackrel{\to }{x}$

## Step 2: Compute the matrix

The matrix is,

$T\stackrel{\to }{\left(x\right)}=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\left[\begin{array}{c}{x}_{1}\\ {x}_{2}\end{array}\right]\phantom{\rule{0ex}{0ex}}T\stackrel{\to }{\left(x\right)}=\left[\begin{array}{c}{x}_{2}\\ {x}_{1}\end{array}\right]$

## Step 3: Graph the matrix

Now, graph the obtained vectors on the given circular face.

Hence, $T\stackrel{\to }{\left(x\right)}$ is obtained by rotating the vector $\stackrel{\to }{x}$ through an angle of $90°$ in the clockwise direction.