CPM Homework Help : PC Problem 12-61

If $l$ and $m$ are parallel lines and $T$ is a linear transformation, why are $lT$ and $mT$ also parallel lines?

Hint:

Slope is the ratio of the change in y to the change in $x$ .
Given any linear transformation, what can you conclude about this ratio?

To convince yourself, try using a transformation with a set of parallel lines. What do you notice?

Example:

$l= \langle t, 3t \rangle$

$m = \langle t, 3t + 1\rangle$

$T = \left[ \begin{array} { l } { 1 } & { 2 } \\ { 3 } & { 4 } \end{array} \right]$

$lT=\langle10t, 14t \rangle$

$mt=\langle 10t+3, 14t+4 \rangle$

$y=\frac{14}{10}x$

$y=\frac{14}{10}(x - 3)+4$