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

Answer:

$\left[\begin{array}{cccc} 1 & -3 & 5 & -4 \\ 1 & -2 & 4 & -3 \end{array}\right]$
$T\left( \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -6 \end{array}\right] \right)= \left[\begin{array}{c} -1 \\ -4 \end{array}\right]$