One point on the line is $\left(0, 2\right)$. Its image is:

$( 0,2 ) \left[ \begin{array} { c c } { 2 } & { 1 } \\ { - 1 } & { 1 } \end{array} \right] = ( - 2,2 )$

Repeat this process for another point on the original line.

Given the points $\left(−2, 2\right)$ and $\left(a, b\right)$, the vector equation of the line is $\lt −2, 2 \gt + \lt a, b \gt t$.