1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} -x + 4 \, y + 2 \, z \\ 2 \, x - 8 \, y - 5 \, z \end{array}\right] .\]

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

    \[ \left[\begin{array}{ccc} 0 & -2 & -5 \\ -1 & 6 & 4 \\ 0 & -5 & -4 \\ -1 & 3 & -2 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -6 \\ 5 \\ 4 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} -1 & 4 & 2 \\ 2 & -8 & -5 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -6 \\ 5 \\ 4 \end{array}\right] \right)= \left[\begin{array}{c} -30 \\ 52 \\ -41 \\ 13 \end{array}\right] \]