Chapter 2.4, Problem 7CFU

To determine

To find: the coordinate of the square using matrices transformation and graph the pre image and image on same coordinate.

Answer to Problem 7CFU

The image vertices used to define the reflection are:

  $\left(1,2\right),\left(4,1\right),\left(3,-2\right)\text{ and }\left(0,-1\right)$ .

Explanation of Solution

Given:

Square $ABCD$ has vertices at $\left(-1,2\right),\left(-4,1\right),\left(-3,-2\right)\text{ and }\left(0,-1\right)$ .

Concept used:

The image of linear transformation or matrix is the span of the vector of the linear transformation.

It can be written as $Im\left(A\right)$ .

For example, consider the matrix $A$ which is equal to:

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

Multiplying this by $2\times 1$ gives a $3\times 1$ matrix. However, regardless of what vector is chosen to multiply by, there are some vectors that can’t be the result.

Result is not image of $A$ . The vector that are possible belong to the span of $A$ . In this case, the span be represented by a parametrized matrix, where $t\text{ and }s$ can be any number:

  $\left[\begin{array}{cc}1& 0\\ 0& 2\\ 0& 1\end{array}\right]\times \left[\begin{array}{c}s\\ t\end{array}\right]=\left[\begin{array}{c}s\\ 2t\\ t\end{array}\right]=Im\left(A\right)$ .

Matrices multiplication:

  $\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{cc}e& f\\ g& h\end{array}\right]=\left[\begin{array}{cc}ae+ag& af+bh\\ ce+dg& cf+dh\end{array}\right].$

Calculation:

Square $ABCD$ has vertices at $\left(-1,2\right),\left(-4,1\right),\left(-3,-2\right)\text{ and }\left(0,-1\right)$ .

First write the vertex matrix for the points used to define a square.

  $\begin{array}{c}x-\text{coordinate}\\ y-\text{coordinate}\end{array}\left[\begin{array}{cccc}\begin{array}{l}A\\ -1\end{array}& \begin{array}{l}B\\ -4\end{array}& \begin{array}{l}C\\ -3\end{array}& \begin{array}{l}D\\ 0\end{array}\\ 2& 1& -2& -1\end{array}\right]$

Reflection over the $y$ axis multiply the vertex matrix $\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ .

  $\begin{array}{l}\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{cccc}-1& -4& -3& 0\\ 2& 1& -2& -1\end{array}\right].\\ \Rightarrow \left[\begin{array}{cccc}1+0& 4+0& 3+0& 0+0\\ 0+2& 0+1& 0-2& 0-1\end{array}\right].\\ \Rightarrow \left[\begin{array}{cccc}\begin{array}{l}{A}^{\prime }\\ 1\end{array}& \begin{array}{l}{B}^{\prime }\\ 4\end{array}& \begin{array}{l}{C}^{\prime }\\ 3\end{array}& \begin{array}{l}{D}^{\prime }\\ 0\end{array}\\ 2& 1& -2& -1\end{array}\right].\end{array}$

The image vertices used to define the reflection are:

  $\left(1,2\right),\left(4,1\right),\left(3,-2\right)\text{ and }\left(0,-1\right)$ .

The graph of the two squares:

Hence, the image vertices used to define the reflection are: $\left(1,2\right),\left(4,1\right),\left(3,-2\right)\text{ and }\left(0,-1\right)$ .