1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 4 \) given by

    \[S\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right) = \left[\begin{array}{c} 4 \, x_{1} + x_{2} + 8 \, x_{3} \\ -x_{1} - 2 \, x_{3} \\ x_{1} + 3 \, x_{2} + 3 \, x_{3} \\ -2 \, x_{1} + 3 \, x_{2} - 8 \, x_{3} \end{array}\right] .\]

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

    \[ \left[\begin{array}{cccc} -1 & -5 & -1 & -4 \\ -2 & -5 & 2 & -7 \\ 1 & 4 & 0 & 4 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -3 \\ 7 \\ -1 \\ 0 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} 4 & 1 & 8 \\ -1 & 0 & -2 \\ 1 & 3 & 3 \\ -2 & 3 & -8 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -3 \\ 7 \\ -1 \\ 0 \end{array}\right] \right)= \left[\begin{array}{c} -31 \\ -31 \\ 25 \end{array}\right] \]