Chapter 11.FOM, Problem 1P

The gray square in Table $1$ has the following vertices:

$\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 1\end{array}\right],\left[\begin{array}{c}0\\ 1\end{array}\right]$

Apply each of the three transformations given in Table $1$ to these vertices and sketch the result to verify that each transformation has the indicated effect. Use $c=2$ in the expansion matrix and $c=1$ in the shear matrix.

To determine

To verify:

The three transformations of the gray square have indicated effect.

Answer to Problem 1P

Solution:

It is verified that each transformation has the indicated effect.

Explanation of Solution

Given:

The vertices of the gray square are:

$\left[\begin{array}{c}0\\ 0\end{array}\right]$, $\left[\begin{array}{c}1\\ 0\end{array}\right]$, $\left[\begin{array}{c}1\\ 1\end{array}\right]$, and $\left[\begin{array}{c}0\\ 1\end{array}\right]$.

$c=2$ in expansion matrix and $c=1$ in the shear matrix.

Approach:

Multiplication of Matrices:

If $A$ is an $m\times n$ matrix and $B$ is a $n\times p$ matrix, then their product $AB$ is the $m\times p$ matrix whose $ij-\text{element}$ is the inner product of the $i\text{th}$ row of $A$ and $j\text{th}$ column of $B$.

Transformation matrix:

$•$ Reflection in $x-\text{axis}$

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

$•$ Expansion in $x-\text{direction}$

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

$•$ Shear in $x-\text{direction}$

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

Calculation:

The data matrix is given by

$D=\left[\begin{array}{cccc}0& 1& 1& 0\\ 0& 0& 1& 1\end{array}\right]$

Plot the points on the graph and connect them by a line segment.

Figure $1$

Multiply the $D$ matrix by appropriate transformation matrix.

$\left(1\right)$ Reflection in $x-\text{axis}$: Apply reflection transformation

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

The following matrix is obtained

$\begin{array}{rll}TD& =& \left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]\left[\begin{array}{cccc}0& 1& 1& 0\\ 0& 0& 1& 1\end{array}\right]\\ & =& \left[\begin{array}{cccc}0& 1& 1& 0\\ 0& 0& -1& -1\end{array}\right]\end{array}$

Obtain the resulting graph by plotting the points and joining them by a line.

Figure $2$

$\left(2\right)$ Expansion in $x-\text{direction}$

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

Substitute $2$ for $c$ in above matrix.

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

The following matrix is obtained

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

Obtain the resulting graph by plotting the points and joining them by a line.

Figure $3$

$\left(3\right)$ Shear in $x-\text{direction}$

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

Substitute $1$ for $c$ in above matrix.

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

The following matrix is obtained

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

Obtain the resulting graph by plotting the points and joining them by a line.

Figure $4$

Conclusion:

Hence, it is verified that each transformation has the indicated effect.