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Q28E

Expert-verifiedFound in: Page 54

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${\mathit{T}}\overrightarrow{\left(x\right)}{\mathbf{=}}{\mathit{A}}\overrightarrow{\mathbf{x}}$on this face.**

28. ${\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]}$

By a factor of$2$, the face gets scaled along y-axis.

The required graph is,

Let the matrix be,

$A=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]$

Then,

localid="1659703886095" $T\overrightarrow{\left(\mathrm{x}\right)}=A\overrightarrow{\mathrm{x}}\phantom{\rule{0ex}{0ex}}T\overrightarrow{\left(\mathrm{x}\right)}=\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]\overrightarrow{\mathrm{x}}$

The matrix is,

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

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

Hence, $T\overrightarrow{\left(\mathrm{x}\right)}$ is obtained by scaling the vector$\overrightarrow{\mathrm{x}}$ by a factor of $2$ .

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