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

    \[S\left( \left[\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right) = \left[\begin{array}{c} 4 \, x_{1} + x_{2} - 7 \, x_{3} - 6 \, x_{4} \\ 3 \, x_{1} + x_{2} - 5 \, x_{3} - 4 \, x_{4} \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} 1 \\ -2 \\ 7 \\ -6 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cccc} 4 & 1 & -7 & -6 \\ 3 & 1 & -5 & -4 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 1 \\ -2 \\ 7 \\ -6 \end{array}\right] \right)= \left[\begin{array}{c} -30 \end{array}\right] \]