Find the standard matrix for the linear transformation $S:\mathbb{R}^ 3 \to \mathbb{R}^ 3$ given by $S\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} x_{2} + x_{3} \\ -x_{1} - 5 \, x_{3} \\ -x_{1} - 3 \, x_{2} - 7 \, x_{3} \end{array}\right] .$
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 $T\left( \left[\begin{array}{c} 8 \\ 8 \\ 8 \\ 1 \end{array}\right] \right)$ .

Answer:

$\left[\begin{array}{ccc} 0 & 1 & 1 \\ -1 & 0 & -5 \\ -1 & -3 & -7 \end{array}\right]$
$T\left( \left[\begin{array}{c} 8 \\ 8 \\ 8 \\ 1 \end{array}\right] \right)= \left[\begin{array}{c} 33 \end{array}\right]$