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

    \[S\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right) = \left[\begin{array}{c} x_{1} + 2 \, x_{2} - 4 \, x_{3} - x_{4} \\ -2 \, x_{1} - 3 \, x_{2} + 6 \, x_{3} \end{array}\right] .\]

  2. Let \(T:\mathbb{R}^ 4 \to \mathbb{R}^ 1 \) be the linear transformation given by the standard matrix

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

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

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -7 \\ -7 \\ -2 \\ 2 \end{array}\right] \right)= \left[\begin{array}{c} -21 \end{array}\right] \]