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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} 2 \, x - 3 \, y + z + 6 \, {w} \\ x - y + 2 \, {w} \end{array}\right] .\]

  2. Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 3 \) be the linear transformation given by the standard matrix

    \[ \left[\begin{array}{cccc} 3 & 0 & 4 & 5 \\ 0 & 1 & 1 & 3 \\ -1 & -1 & -2 & -4 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 5 \\ 8 \\ 7 \\ -2 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cccc} 2 & -3 & 1 & 6 \\ 1 & -1 & 0 & 2 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 5 \\ 8 \\ 7 \\ -2 \end{array}\right] \right)= \left[\begin{array}{c} 33 \\ 9 \\ -19 \end{array}\right] \]