Science:Math Exam Resources/Courses/MATH152/April 2011/Question A 13/Solution 1

As per the hint, we only have to consider the transformation of 4 special points. These points are the corners of the square. Linear transformations cannot distort the edges of a figure in such a way that they split an edge into a new corner. Therefore, the transformed figure must have the same amount of corners as the original figure. Because of this, it is easiest to just transform the corners and then connect them to make the new figure. Our transformation matrix is

$A={\begin{bmatrix}1&2\\2&1\end{bmatrix}}$

and the four corners of the square are [0,0], [1,0], [0,1], and [1,1]. Firstly notice that the origin will always map to itself because no matrix can multiply [0,0] to produce anything but [0,0]. Therefore, we really only have to transform 3 points. The transformation of [1,0] is,

$A{\begin{bmatrix}1\\0\end{bmatrix}}={\begin{bmatrix}1&2\\2&1\end{bmatrix}}{\begin{bmatrix}1\\0\end{bmatrix}}={\begin{bmatrix}1\\2\end{bmatrix}}.$

The transformation of [0,1] is,

$A{\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}1&2\\2&1\end{bmatrix}}{\begin{bmatrix}0\\1\end{bmatrix}}={\begin{bmatrix}2\\1\end{bmatrix}}$

and the transformation of [1,1] is

$A{\begin{bmatrix}1\\1\end{bmatrix}}={\begin{bmatrix}1&2\\2&1\end{bmatrix}}{\begin{bmatrix}1\\1\end{bmatrix}}={\begin{bmatrix}3\\3\end{bmatrix}}.$

Therefore we have ${\color {blue}[0,0]}$ transforms to ${\color {red}[0,0]}$, ${\color {blue}[1,0]}$ transforms to ${\color {red}[1,2]}$, ${\color {blue}[0,1]}$ transforms to ${\color {red}[2,1]}$ and ${\color {blue}[1,1]}$ transforms to ${\color {red}[3,3]}$. The plot to the right

shows the original points in ${\color {blue}{\textrm {blue}}}$ while the transformed points are in ${\color {red}{\textrm {red}}}$. We complete the figure by connecting the corners.