CPM Homework Help : PC Problem 12-53

Find the images of the four points $\left(0, 0\right)$ , $\left(1, 0\right)$ , $\left(1, 1\right)$ , and $\left(0, 1\right)$ by the matrix $\left[ \begin{array} { c c } { 2 } & { 1 } \\ { - 1 } & { 1 } \end{array} \right]$ .
Step (a):
$(1,0)\begin{bmatrix}2&1\\-1&1\end{bmatrix}=(1(2) + 0 (-1), 1(1) + 0(1))$
Graph these points.
What is the image of the square whose vertices are the points $\left(0, 0\right)$ , $\left(1, 0\right)$ , $\left(1, 1\right)$ , and $\left(0, 1\right)$ ? Shade it on the graph you made in part (b).
What is the name of this geometric figure?
Sketch the images of the lines $y = 0$ , $y = 1$ , $x = 0$ and $x = 1$ .

Steps:

For the line $y = 1$ , all points are of the form $\left(a, 1\right)$ .

$(a,1)\begin{bmatrix}2&1\\-1&1\end{bmatrix}=(2a-1,2a+1)$

Hence $x = 2a − 1$ and $y = a + 1$ .
Write an equation relating x and y.

$y=\frac{1}{2}x+\frac{3}{2}$

Hint:

If you graphed the original line and its image, how would it relate to the original square and its image in parts (a) and (b)?