Chapter 7.2, Problem 60E

Solution

To prove: The image of a point under the given transformation of the plane can be obtained by matrix multiplication.

Given Information:

The given statement is “A vertical stretch or shrink by a factor of $a$ ”.

Explanation:

Consider the given information,

Suppose the point is $\left[\begin{array}{cc}x& y\end{array}\right]$ and the matrix is $\left[\begin{array}{cc}m& n\\ p& q\end{array}\right]$ . So the point $\left[\begin{array}{cc}x& y\end{array}\right]$ is reflected into $\left[\begin{array}{cc}x& ay\end{array}\right]$ for the vertically shrink by a factor a . So, the multiplication is defined as,

  $\left[\begin{array}{cc}x& y\end{array}\right]\left[\begin{array}{cc}m& n\\ p& q\end{array}\right]=\left[\begin{array}{cc}x& ay\end{array}\right]$

Multiply the matrix and write the equation form.

  $\begin{array}{c}mx+py=x\\ nx+qy=ay\end{array}$

Rewrite the obtained equations.

  $\begin{array}{c}mx+py-x=0\\ \left(m-1\right)x+py=0\end{array}$

And,

  $\begin{array}{c}nx+qy-ay=0\\ nx+y\left(q-a\right)=0\end{array}$

For the system of equations to be true for any values of  x , y , then the obtained result are defined as:

  $\begin{array}{c}m-1=0\\ m=1\end{array}$

And,

  $\begin{array}{l}p=0\\ n=0\end{array}$

And,

  $\begin{array}{c}q-a=0\\ q=a\end{array}$

The obtained matrix is $\left[\begin{array}{cc}1& 0\\ 0& a\end{array}\right]$ .

Therefore, the given statement has proved.