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

    \[S\left( \left[\begin{array}{c} x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x - 5 \, y \\ -x + 6 \, y \\ 0 \\ x - 2 \, y \end{array}\right] .\]

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

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

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

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -2 \\ 8 \end{array}\right] \right)= \left[\begin{array}{c} 2 \\ -30 \\ 36 \\ 2 \end{array}\right] \]