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

Answer:

$\left[\begin{array}{cccc} -2 & 1 & 0 & -2 \\ 1 & -1 & 0 & 0 \end{array}\right]$
$T\left( \left[\begin{array}{c} 6 \\ -3 \\ 5 \\ -7 \end{array}\right] \right)= \left[\begin{array}{c} 6 \end{array}\right]$