1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 4 \to \mathbb{R}^ 2 \) given by

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} z \\ -x - 3 \, y + 6 \, z + 3 \, {w} \end{array}\right] .\]

  2. Let \(T:\mathbb{R}^ 3 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix

    \[ \left[\begin{array}{ccc} -5 & -2 & -3 \\ -2 & -1 & -2 \\ 0 & -2 & -7 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 3 \\ -4 \\ -6 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ -1 & -3 & 6 & 3 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 3 \\ -4 \\ -6 \end{array}\right] \right)= \left[\begin{array}{c} 11 \\ 10 \\ 50 \end{array}\right] \]