Find the standard matrix for the linear transformation $S:\mathbb{R}^ 2 \to \mathbb{R}^ 4$ given by $S\left( \left[\begin{array}{c} x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x - y \\ 3 \, x - 2 \, y \\ -3 \, x + 3 \, y \\ -x - y \end{array}\right] .$
Let $T:\mathbb{R}^ 3 \to \mathbb{R}^ 4$ be the linear transformation given by the standard matrix $\left[\begin{array}{ccc} -1 & -1 & 0 \\ -2 & -3 & -7 \\ -2 & -3 & -6 \\ 1 & 1 & 5 \end{array}\right] .$ Compute $T\left( \left[\begin{array}{c} 7 \\ 8 \\ 0 \end{array}\right] \right)$ .

Answer:

$\left[\begin{array}{cc} 1 & -1 \\ 3 & -2 \\ -3 & 3 \\ -1 & -1 \end{array}\right]$
$T\left( \left[\begin{array}{c} 7 \\ 8 \\ 0 \end{array}\right] \right)= \left[\begin{array}{c} -15 \\ -38 \\ -38 \\ 15 \end{array}\right]$