A2 - Standard matrices


Example 1 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} x + y \\ x + 2 \, y + 3 \, z \\ x + 3 \, y + 7 \, z \\ x - 2 \, z \end{array}\right] .\]

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

    \[ \left[\begin{array}{cccc} 1 & 3 & -1 & 8 \\ -1 & -2 & 0 & -6 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 1 \\ -7 \\ 0 \\ 6 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} 1 & 1 & 0 \\ 1 & 2 & 3 \\ 1 & 3 & 7 \\ 1 & 0 & -2 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 1 \\ -7 \\ 0 \\ 6 \end{array}\right] \right)= \left[\begin{array}{c} 28 \\ -23 \end{array}\right] \]


Example 2 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} 8 \\ -5 \\ -2 \\ -8 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} 1 & 2 & -6 \\ 0 & 1 & -3 \\ 4 & 2 & -5 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 8 \\ -5 \\ -2 \\ -8 \end{array}\right] \right)= \left[\begin{array}{c} 35 \end{array}\right] \]


Example 3 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

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

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

    \[ \left[\begin{array}{cc} 7 & 1 \\ 2 & 3 \\ 3 & 0 \\ -3 & -5 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 8 \\ -5 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 8 \\ -5 \end{array}\right] \right)= \left[\begin{array}{c} 51 \\ 1 \\ 24 \\ 1 \end{array}\right] \]


Example 4 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} x + 2 \, y + z \\ -4 \, x - 7 \, y - 5 \, z \end{array}\right] .\]

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

    \[ \left[\begin{array}{cccc} 1 & 2 & -3 & -5 \\ 1 & 3 & -4 & -7 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -5 \\ -8 \\ -2 \\ -2 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -5 \\ -8 \\ -2 \\ -2 \end{array}\right] \right)= \left[\begin{array}{c} -5 \\ -7 \end{array}\right] \]


Example 5 πŸ”—

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

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

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

    \[ \left[\begin{array}{cccc} 1 & -1 & 2 & 0 \\ 1 & 0 & 5 & 3 \\ -1 & 3 & 5 & 7 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ 3 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ 3 \end{array}\right] \right)= \left[\begin{array}{c} 6 \\ 28 \\ 45 \end{array}\right] \]


Example 6 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) given by

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

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

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

    Compute \(T\left( \left[\begin{array}{c} 1 \\ 5 \\ -8 \\ -8 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 1 \\ 5 \\ -8 \\ -8 \end{array}\right] \right)= \left[\begin{array}{c} 144 \\ 45 \end{array}\right] \]


Example 7 πŸ”—

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

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

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

    \[ \left[\begin{array}{ccc} 0 & 1 & 5 \\ -1 & 0 & 1 \\ -3 & -2 & -6 \end{array}\right] .\]

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

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -3 \\ 7 \\ 7 \end{array}\right] \right)= \left[\begin{array}{c} 42 \\ 10 \\ -47 \end{array}\right] \]


Example 8 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ -1 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} 2 & 1 & 6 \\ -1 & 0 & -2 \\ 3 & -1 & 5 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ -1 \end{array}\right] \right)= \left[\begin{array}{c} 13 \\ -16 \\ -4 \end{array}\right] \]


Example 9 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

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

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

    \[ \left[\begin{array}{cccc} 1 & -1 & 1 & 4 \\ 2 & -1 & 3 & 8 \\ -2 & 2 & -1 & -7 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 0 \\ -6 \\ -7 \\ 7 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} 4 & -3 & -7 \\ 3 & -2 & -5 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 0 \\ -6 \\ -7 \\ 7 \end{array}\right] \right)= \left[\begin{array}{c} 27 \\ 41 \\ -54 \end{array}\right] \]


Example 10 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} 4 \\ 5 \\ 2 \\ 3 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} -4 & 1 & 1 \\ 2 & 1 & -7 \\ 0 & -1 & 6 \\ 5 & -1 & -2 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 4 \\ 5 \\ 2 \\ 3 \end{array}\right] \right)= \left[\begin{array}{c} 20 \end{array}\right] \]


Example 11 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} x + 2 \, y - 3 \, z + 3 \, {w} \end{array}\right] .\]

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

    \[ \left[\begin{array}{cc} -2 & -7 \\ 1 & 3 \\ 0 & 1 \end{array}\right] .\]

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

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 0 \\ -7 \end{array}\right] \right)= \left[\begin{array}{c} 49 \\ -21 \\ -7 \end{array}\right] \]


Example 12 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} x + 3 \, {w} \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} -3 \\ 1 \\ -6 \\ -7 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -3 \\ 1 \\ -6 \\ -7 \end{array}\right] \right)= \left[\begin{array}{c} 0 \end{array}\right] \]


Example 13 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

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

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

    \[ \left[\begin{array}{ccc} -2 & 0 & -5 \\ -2 & 1 & -6 \\ 3 & 0 & 7 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 1 \\ -8 \\ 4 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} -1 & 3 & -7 \\ -1 & 2 & -5 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 1 \\ -8 \\ 4 \end{array}\right] \right)= \left[\begin{array}{c} -22 \\ -34 \\ 31 \end{array}\right] \]


Example 14 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} x + y - 3 \, z + {w} \end{array}\right] .\]

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

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

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

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -2 \\ 8 \end{array}\right] \right)= \left[\begin{array}{c} -38 \\ -10 \\ -66 \\ 58 \end{array}\right] \]


Example 15 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} -3 \, x - 3 \, y - 2 \, z + {w} \\ 4 \, x + 5 \, y + 2 \, z \\ 2 \, x + 2 \, y + z \end{array}\right] .\]

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

    \[ \left[\begin{array}{cc} 1 & 1 \\ 4 & 5 \\ 2 & 3 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -5 \\ -4 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cccc} -3 & -3 & -2 & 1 \\ 4 & 5 & 2 & 0 \\ 2 & 2 & 1 & 0 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -5 \\ -4 \end{array}\right] \right)= \left[\begin{array}{c} -9 \\ -40 \\ -22 \end{array}\right] \]


Example 16 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} x + 2 \, z - 4 \, {w} \\ y - 5 \, z + 7 \, {w} \\ y - 4 \, z + 6 \, {w} \end{array}\right] .\]

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

    \[ \left[\begin{array}{ccc} -3 & 6 & -2 \\ -4 & 8 & -3 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -7 \\ 4 \\ 0 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cccc} 1 & 0 & 2 & -4 \\ 0 & 1 & -5 & 7 \\ 0 & 1 & -4 & 6 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -7 \\ 4 \\ 0 \end{array}\right] \right)= \left[\begin{array}{c} 45 \\ 60 \end{array}\right] \]


Example 17 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) given by

    \[S\left( \left[\begin{array}{c} x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x + 2 \, y \\ -2 \, x - 3 \, y \\ 5 \, y \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} -8 \\ 7 \\ 2 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cc} 1 & 2 \\ -2 & -3 \\ 0 & 5 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -8 \\ 7 \\ 2 \end{array}\right] \right)= \left[\begin{array}{c} -1 \\ 0 \\ 2 \\ 8 \end{array}\right] \]


Example 18 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} -2 \\ -8 \\ 8 \\ -4 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -2 \\ -8 \\ 8 \\ -4 \end{array}\right] \right)= \left[\begin{array}{c} -2 \\ -4 \end{array}\right] \]


Example 19 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} x + 5 \, y + 3 \, z - 6 \, {w} \\ -x - 5 \, y - 2 \, z + 6 \, {w} \\ {w} \end{array}\right] .\]

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

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

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

Answer:

  1. \[ \left[\begin{array}{cccc} 1 & 5 & 3 & -6 \\ -1 & -5 & -2 & 6 \\ 0 & 0 & 0 & 1 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -7 \\ -1 \end{array}\right] \right)= \left[\begin{array}{c} -33 \\ -1 \\ 14 \end{array}\right] \]


Example 20 πŸ”—

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

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

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

    \[ \left[\begin{array}{cccc} 5 & -7 & 3 & -4 \\ -2 & 3 & -1 & 2 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -1 \\ -6 \\ 3 \\ 8 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -1 \\ -6 \\ 3 \\ 8 \end{array}\right] \right)= \left[\begin{array}{c} 14 \\ -3 \end{array}\right] \]


Example 21 πŸ”—

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

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

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

    \[ \left[\begin{array}{cc} -4 & -4 \\ 2 & 5 \\ 0 & 5 \\ -5 & -6 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -7 \\ 5 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} -7 & -2 & -6 \\ 4 & 1 & 4 \\ -5 & -1 & -5 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -7 \\ 5 \end{array}\right] \right)= \left[\begin{array}{c} 8 \\ 11 \\ 25 \\ 5 \end{array}\right] \]


Example 22 πŸ”—

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

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

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

    \[ \left[\begin{array}{ccc} -1 & 0 & -2 \\ -1 & 4 & -7 \\ -2 & 3 & -8 \end{array}\right] .\]

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

Answer:

  1. \[ \left[\begin{array}{cc} 0 & -2 \\ 4 & -7 \\ 1 & 0 \\ 1 & 3 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 2 \\ 1 \\ -2 \end{array}\right] \right)= \left[\begin{array}{c} 2 \\ 16 \\ 15 \end{array}\right] \]


Example 23 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} 3 \\ 6 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 3 \\ 6 \end{array}\right] \right)= \left[\begin{array}{c} -27 \\ 6 \\ -30 \\ 30 \end{array}\right] \]


Example 24 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} 3 \, x + 8 \, y - 3 \, z \\ -2 \, x - 5 \, y + 2 \, z \end{array}\right] .\]

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

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

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

Answer:

  1. \[ \left[\begin{array}{ccc} 3 & 8 & -3 \\ -2 & -5 & 2 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -8 \\ -8 \end{array}\right] \right)= \left[\begin{array}{c} 32 \\ -40 \\ 32 \\ 0 \end{array}\right] \]


Example 25 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

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

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

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

    Compute \(T\left( \left[\begin{array}{c} -1 \\ 3 \\ -7 \\ -6 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -1 \\ 3 \\ -7 \\ -6 \end{array}\right] \right)= \left[\begin{array}{c} -27 \end{array}\right] \]


Example 26 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} -z + 2 \, {w} \\ x + 3 \, y - 2 \, z + 5 \, {w} \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} 7 \\ -8 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cccc} 0 & 0 & -1 & 2 \\ 1 & 3 & -2 & 5 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 7 \\ -8 \end{array}\right] \right)= \left[\begin{array}{c} 47 \\ 39 \\ -54 \end{array}\right] \]


Example 27 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) given by

    \[S\left( \left[\begin{array}{c} x_{1} \\ x_{2} \end{array}\right] \right) = \left[\begin{array}{c} -x_{1} - 4 \, x_{2} \\ 2 \, x_{1} + 7 \, x_{2} \\ -2 \, x_{1} - 8 \, x_{2} \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} -6 \\ 0 \\ 0 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cc} -1 & -4 \\ 2 & 7 \\ -2 & -8 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -6 \\ 0 \\ 0 \end{array}\right] \right)= \left[\begin{array}{c} -6 \\ 24 \end{array}\right] \]


Example 28 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) given by

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

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

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

    Compute \(T\left( \left[\begin{array}{c} -5 \\ 1 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -5 \\ 1 \end{array}\right] \right)= \left[\begin{array}{c} -1 \\ 1 \\ -1 \end{array}\right] \]


Example 29 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) given by

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

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

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

    Compute \(T\left( \left[\begin{array}{c} 2 \\ 6 \\ -5 \\ -8 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 2 \\ 6 \\ -5 \\ -8 \end{array}\right] \right)= \left[\begin{array}{c} 16 \end{array}\right] \]


Example 30 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

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

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

    \[ \left[\begin{array}{cccc} 1 & 5 & 0 & -3 \\ 0 & 1 & 1 & 1 \\ 1 & 8 & 4 & 2 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -5 \\ -3 \\ -2 \\ -5 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -5 \\ -3 \\ -2 \\ -5 \end{array}\right] \right)= \left[\begin{array}{c} -5 \\ -10 \\ -47 \end{array}\right] \]


Example 31 πŸ”—

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

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

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

    \[ \left[\begin{array}{cc} 0 & 3 \\ 1 & -3 \\ 1 & -8 \\ 1 & -7 \end{array}\right] .\]

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

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 2 \\ -6 \end{array}\right] \right)= \left[\begin{array}{c} -18 \\ 20 \\ 50 \\ 44 \end{array}\right] \]


Example 32 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x - y \\ 2 \, x - y \\ 5 \, x \\ -5 \, x + 7 \, y \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} 4 \\ 8 \\ 3 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 4 \\ 8 \\ 3 \end{array}\right] \right)= \left[\begin{array}{c} -18 \\ -7 \\ 3 \\ -12 \end{array}\right] \]


Example 33 πŸ”—

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

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

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

    \[ \left[\begin{array}{ccc} 4 & 6 & 7 \\ 1 & 2 & 3 \\ -3 & -5 & -6 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 6 \\ -7 \\ 4 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 6 \\ -7 \\ 4 \end{array}\right] \right)= \left[\begin{array}{c} 10 \\ 4 \\ -7 \end{array}\right] \]


Example 34 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} x + 4 \, y - 2 \, z - 6 \, {w} \\ z + 2 \, {w} \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} 0 \\ -2 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 0 \\ -2 \end{array}\right] \right)= \left[\begin{array}{c} 6 \\ -8 \\ 12 \end{array}\right] \]


Example 35 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) given by

    \[S\left( \left[\begin{array}{c} x \\ y \end{array}\right] \right) = \left[\begin{array}{c} -x + 6 \, y \\ 4 \, x - 7 \, y \\ x - 4 \, y \end{array}\right] .\]

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

    \[ \left[\begin{array}{ccc} 1 & 2 & -3 \\ -2 & -3 & 7 \\ 0 & 0 & 1 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 3 \\ 2 \\ 4 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 3 \\ 2 \\ 4 \end{array}\right] \right)= \left[\begin{array}{c} -5 \\ 16 \\ 4 \end{array}\right] \]


Example 36 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 2 \to \mathbb{R}^ 3 \) given by

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

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

    \[ \left[\begin{array}{cc} 4 & 7 \\ -1 & -2 \\ -3 & -5 \\ 3 & 4 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -3 \\ -6 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -3 \\ -6 \end{array}\right] \right)= \left[\begin{array}{c} -54 \\ 15 \\ 39 \\ -33 \end{array}\right] \]


Example 37 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} -2 \, x + y - 2 \, {w} \\ -3 \, x + y + z - 4 \, {w} \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} -2 \\ 3 \\ -2 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -2 \\ 3 \\ -2 \end{array}\right] \right)= \left[\begin{array}{c} -1 \\ 10 \\ -13 \end{array}\right] \]


Example 38 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} 5 \\ -7 \\ 0 \\ 8 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 5 \\ -7 \\ 0 \\ 8 \end{array}\right] \right)= \left[\begin{array}{c} 19 \end{array}\right] \]


Example 39 πŸ”—

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

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

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

    \[ \left[\begin{array}{cccc} 2 & 1 & 2 & -6 \\ 2 & 1 & 3 & -7 \\ -1 & -1 & 2 & -1 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -8 \\ 6 \\ 4 \\ 8 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -8 \\ 6 \\ 4 \\ 8 \end{array}\right] \right)= \left[\begin{array}{c} -50 \\ -54 \\ 2 \end{array}\right] \]


Example 40 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} -z + 2 \, {w} \\ x + 4 \, y - 2 \, z + 4 \, {w} \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} 7 \\ -8 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 7 \\ -8 \end{array}\right] \right)= \left[\begin{array}{c} -32 \\ -8 \\ -63 \end{array}\right] \]


Example 41 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} 4 \, x + 5 \, y + 3 \, z - 4 \, {w} \\ 3 \, x - 2 \, y - 4 \, {w} \\ x + 2 \, y + z - {w} \end{array}\right] .\]

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

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

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

Answer:

  1. \[ \left[\begin{array}{cccc} 4 & 5 & 3 & -4 \\ 3 & -2 & 0 & -4 \\ 1 & 2 & 1 & -1 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -7 \\ 1 \\ 6 \end{array}\right] \right)= \left[\begin{array}{c} -45 \\ -37 \end{array}\right] \]


Example 42 πŸ”—

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

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

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

    \[ \left[\begin{array}{ccc} 0 & -3 & -6 \\ 1 & 3 & 1 \\ -3 & -2 & 5 \\ 1 & 5 & 6 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -1 \\ 6 \\ -8 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -1 \\ 6 \\ -8 \end{array}\right] \right)= \left[\begin{array}{c} 30 \\ 9 \\ -49 \\ -19 \end{array}\right] \]


Example 43 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \end{array}\right] \right) = \left[\begin{array}{c} x - 5 \, y \\ x - 4 \, y \\ -x + 7 \, y \\ -x + y \end{array}\right] .\]

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

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

    Compute \(T\left( \left[\begin{array}{c} 3 \\ 1 \\ -2 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 3 \\ 1 \\ -2 \end{array}\right] \right)= \left[\begin{array}{c} 2 \\ -6 \\ -5 \\ -6 \end{array}\right] \]


Example 44 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \\ {w} \end{array}\right] \right) = \left[\begin{array}{c} y + 3 \, z - 2 \, {w} \\ -x + 2 \, y + 7 \, z - 7 \, {w} \end{array}\right] .\]

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

    \[ \left[\begin{array}{cc} 1 & 3 \\ 0 & 1 \\ 0 & -1 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -8 \\ -6 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cccc} 0 & 1 & 3 & -2 \\ -1 & 2 & 7 & -7 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -8 \\ -6 \end{array}\right] \right)= \left[\begin{array}{c} -26 \\ -6 \\ 6 \end{array}\right] \]


Example 45 πŸ”—

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

    \[S\left( \left[\begin{array}{c} x \\ y \\ z \end{array}\right] \right) = \left[\begin{array}{c} -5 \, x - 2 \, y + 5 \, z \\ 3 \, x + y - 4 \, z \\ 3 \, x + y - 3 \, z \end{array}\right] .\]

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

    \[ \left[\begin{array}{cccc} -3 & -7 & 1 & -8 \\ 1 & 2 & -1 & 5 \\ 2 & 4 & -1 & 6 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ 7 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{ccc} -5 & -2 & 5 \\ 3 & 1 & -4 \\ 3 & 1 & -3 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ 7 \end{array}\right] \right)= \left[\begin{array}{c} -23 \\ 25 \\ 22 \end{array}\right] \]


Example 46 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} 0 \\ 4 \\ 4 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cc} 2 & -7 \\ 1 & -4 \\ 1 & -6 \\ -1 & 2 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 0 \\ 4 \\ 4 \end{array}\right] \right)= \left[\begin{array}{c} -12 \\ -8 \\ 0 \end{array}\right] \]


Example 47 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} -5 \\ -1 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cccc} 0 & 3 & -5 & 3 \\ 1 & 0 & -4 & -2 \\ -1 & 4 & -3 & 6 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} -5 \\ -1 \end{array}\right] \right)= \left[\begin{array}{c} -8 \\ -1 \\ 4 \end{array}\right] \]


Example 48 πŸ”—

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

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

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

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

    Compute \(T\left( \left[\begin{array}{c} 4 \\ 1 \end{array}\right] \right)\).

Answer:

  1. \[ \left[\begin{array}{cc} -3 & -2 \\ -2 & -7 \\ -2 & -3 \\ -1 & -3 \end{array}\right] \]

  2. \[T\left( \left[\begin{array}{c} 4 \\ 1 \end{array}\right] \right)= \left[\begin{array}{c} -20 \\ -24 \\ -12 \\ 25 \end{array}\right] \]


Example 49 πŸ”—

  1. Find the standard matrix for the linear transformation \(S:\mathbb{R}^ 3 \to \mathbb{R}^ 2 \) given by

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

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

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

    Compute \(T\left( \left[\begin{array}{c} -7 \\ 0 \\ 8 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} -7 \\ 0 \\ 8 \end{array}\right] \right)= \left[\begin{array}{c} -1 \\ -2 \end{array}\right] \]


Example 50 πŸ”—

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

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

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

    \[ \left[\begin{array}{cc} 3 & 7 \\ -2 & -1 \\ 2 & 6 \end{array}\right] .\]

    Compute \(T\left( \left[\begin{array}{c} 1 \\ 4 \end{array}\right] \right)\).

Answer:

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

  2. \[T\left( \left[\begin{array}{c} 1 \\ 4 \end{array}\right] \right)= \left[\begin{array}{c} 31 \\ -6 \\ 26 \end{array}\right] \]