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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} -5 \\ 3 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -5 \\ 3 \end{array}\right] \right)= \left[\begin{array}{c} 34 \\ -21 \\ 6 \end{array}\right] \]