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

Answer:

$\left[\begin{array}{ccc} -1 & 3 & 4 \\ -1 & 0 & -1 \\ 2 & -2 & -1 \\ -3 & 1 & -6 \end{array}\right]$
$T\left( \left[\begin{array}{c} -3 \\ -7 \\ 6 \\ -5 \end{array}\right] \right)= \left[\begin{array}{c} -27 \\ -8 \end{array}\right]$