A2 - Linear maps and matrices


Example 1

A2 - Linear maps and matrices (ver. 1)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -3 \\ 8 \\ 1 \\ -5 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 4 \, x_{2} - 3 \, x_{3} + 5 \, x_{4} \\ x_{2} + x_{3} - 2 \, x_{4} \end{array}\right]\]

\[\left[\begin{array}{r} -63 \\ 19 \end{array}\right]\]


Example 2

A2 - Linear maps and matrices (ver. 2)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -8 \\ -6 \\ 5 \\ -6 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + 5 \, x_{2} + 7 \, x_{3} - 5 \, x_{4} \\ x_{2} + 2 \, x_{3} - x_{4} \end{array}\right]\]

\[\left[\begin{array}{r} 27 \\ 10 \end{array}\right]\]


Example 3

A2 - Linear maps and matrices (ver. 3)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} 3 \, x_{2} + 8 \, x_{3} \\ x_{2} + 3 \, x_{3} \\ x_{1} - x_{2} - 2 \, x_{3} \\ -2 \, x_{1} - 3 \, x_{2} - 6 \, x_{3} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -6 \\ 5 \\ 1 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 23 \\ 8 \\ -13 \\ -9 \end{array}\right]\]


Example 4

A2 - Linear maps and matrices (ver. 4)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 2 \\ 7 \\ 2 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 11 \\ -26 \end{array}\right]\]


Example 5

A2 - Linear maps and matrices (ver. 5)

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

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

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -8 \\ -6 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 8 \\ -10 \\ -8 \\ 4 \end{array}\right]\]


Example 6

A2 - Linear maps and matrices (ver. 6)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -6 \\ 6 \\ 7 \\ 3 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + x_{2} - 2 \, x_{3} - x_{4} \\ x_{2} - 5 \, x_{3} - 5 \, x_{4} \\ x_{2} - 4 \, x_{3} - 4 \, x_{4} \end{array}\right]\]

\[\left[\begin{array}{r} -17 \\ -44 \\ -34 \end{array}\right]\]


Example 7

A2 - Linear maps and matrices (ver. 7)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 0 \\ 8 \\ 2 \\ -3 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 5 \\ -4 \\ 2 \end{array}\right]\]


Example 8

A2 - Linear maps and matrices (ver. 8)

Let \(T:\mathbb R^{4}\to\mathbb R^{3}\) be the transformation given by

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} -3 \, x_{1} - 3 \, x_{2} + 7 \, x_{3} + 2 \, x_{4} \\ -4 \, x_{1} - 3 \, x_{2} + 8 \, x_{3} + x_{4} \\ 2 \, x_{1} + 2 \, x_{2} - 5 \, x_{3} - 2 \, x_{4} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -6 \\ -3 \\ -5 \\ 0 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -8 \\ -7 \\ 7 \end{array}\right]\]


Example 9

A2 - Linear maps and matrices (ver. 9)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -3 \\ 1 \\ 7 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 3 \, x_{3} \\ x_{2} - 3 \, x_{3} \end{array}\right]\]

\[\left[\begin{array}{r} -24 \\ -20 \end{array}\right]\]


Example 10

A2 - Linear maps and matrices (ver. 10)

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

\[\left[\begin{array}{rrrr} 1 & 0 & 1 & -2 \\ 5 & 1 & 2 & -8 \end{array}\right]\]

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -7 \\ -6 \\ 0 \\ 1 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + x_{3} - 2 \, x_{4} \\ 5 \, x_{1} + x_{2} + 2 \, x_{3} - 8 \, x_{4} \end{array}\right]\]

\[\left[\begin{array}{r} -9 \\ -49 \end{array}\right]\]


Example 11

A2 - Linear maps and matrices (ver. 11)

Let \(T:\mathbb R^{3}\to\mathbb R^{3}\) be the transformation given by

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

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -7 \\ -5 \\ -1 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -3 \\ -12 \\ -13 \end{array}\right]\]


Example 12

A2 - Linear maps and matrices (ver. 12)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -6 \\ 4 \\ -5 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -13 \\ -1 \\ -23 \end{array}\right]\]


Example 13

A2 - Linear maps and matrices (ver. 13)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -5 \\ 8 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 19 \\ 8 \\ -21 \\ -16 \end{array}\right]\]


Example 14

A2 - Linear maps and matrices (ver. 14)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -5 \\ 2 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} -x_{1} + 3 \, x_{2} \\ x_{1} - 2 \, x_{2} \\ -2 \, x_{1} + 6 \, x_{2} \end{array}\right]\]

\[\left[\begin{array}{r} 11 \\ -9 \\ 22 \end{array}\right]\]


Example 15

A2 - Linear maps and matrices (ver. 15)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -3 \\ -5 \\ 4 \\ -4 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 31 \\ -19 \\ 23 \end{array}\right]\]


Example 16

A2 - Linear maps and matrices (ver. 16)

Let \(T:\mathbb R^{3}\to\mathbb R^{3}\) be the transformation given by

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

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -2 \\ 5 \\ -4 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 3 \\ -8 \\ 7 \end{array}\right]\]


Example 17

A2 - Linear maps and matrices (ver. 17)

Let \(T:\mathbb R^{3}\to\mathbb R^{3}\) be the transformation given by

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

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -3 \\ 5 \\ 5 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 17 \\ -51 \\ -29 \end{array}\right]\]


Example 18

A2 - Linear maps and matrices (ver. 18)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 0 \\ -4 \\ 8 \\ -4 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -16 \\ 8 \end{array}\right]\]


Example 19

A2 - Linear maps and matrices (ver. 19)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 1 \\ 5 \\ 8 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -6 \\ -19 \\ -45 \\ 12 \end{array}\right]\]


Example 20

A2 - Linear maps and matrices (ver. 20)

Let \(T:\mathbb R^{3}\to\mathbb R^{3}\) be the transformation given by

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

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -5 \\ 8 \\ 2 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -41 \\ -27 \\ 16 \end{array}\right]\]


Example 21

A2 - Linear maps and matrices (ver. 21)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -1 \\ -1 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -8 \\ 3 \\ -9 \end{array}\right]\]


Example 22

A2 - Linear maps and matrices (ver. 22)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} -x_{1} - 6 \, x_{2} \\ x_{2} \\ 2 \, x_{1} - 3 \, x_{2} \\ x_{1} + x_{2} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -3 \\ 5 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -27 \\ 5 \\ -21 \\ 2 \end{array}\right]\]


Example 23

A2 - Linear maps and matrices (ver. 23)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -5 \\ -2 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} -4 \, x_{2} \\ x_{1} - x_{2} \\ x_{1} + 2 \, x_{2} \end{array}\right]\]

\[\left[\begin{array}{r} 8 \\ -3 \\ -9 \end{array}\right]\]


Example 24

A2 - Linear maps and matrices (ver. 24)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} 5 \, x_{1} + 3 \, x_{3} \\ -4 \, x_{1} + x_{2} \\ -2 \, x_{1} - x_{3} \\ 5 \, x_{1} + 4 \, x_{3} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -3 \\ 1 \\ -8 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -39 \\ 13 \\ 14 \\ -47 \end{array}\right]\]


Example 25

A2 - Linear maps and matrices (ver. 25)

Let \(T:\mathbb R^{3}\to\mathbb R^{4}\) be the transformation given by

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} -x_{1} + 3 \, x_{2} + 4 \, x_{3} \\ -x_{1} + 4 \, x_{2} - 7 \, x_{3} \\ -2 \, x_{1} + 3 \, x_{2} - 8 \, x_{3} \\ -x_{1} + 3 \, x_{2} - x_{3} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -3 \\ -1 \\ -8 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -32 \\ 55 \\ 67 \\ 8 \end{array}\right]\]


Example 26

A2 - Linear maps and matrices (ver. 26)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -2 \\ 4 \\ 2 \\ 2 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} 2 \, x_{1} - 5 \, x_{2} - 2 \, x_{3} - 7 \, x_{4} \\ x_{1} - 2 \, x_{2} - x_{3} - 3 \, x_{4} \\ -5 \, x_{2} + x_{3} - 3 \, x_{4} \end{array}\right]\]

\[\left[\begin{array}{r} -42 \\ -18 \\ -24 \end{array}\right]\]


Example 27

A2 - Linear maps and matrices (ver. 27)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} -x_{1} - 8 \, x_{2} \\ -3 \, x_{1} + 7 \, x_{2} \\ -x_{1} - 3 \, x_{2} \\ -3 \, x_{1} + 7 \, x_{2} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 3 \\ 4 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -35 \\ 19 \\ -15 \\ 19 \end{array}\right]\]


Example 28

A2 - Linear maps and matrices (ver. 28)

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

\[\left[\begin{array}{rrr} 1 & 1 & -5 \\ 1 & 2 & -8 \\ 2 & 1 & -6 \\ 0 & -1 & 6 \end{array}\right]\]

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 3 \\ -8 \\ -6 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + x_{2} - 5 \, x_{3} \\ x_{1} + 2 \, x_{2} - 8 \, x_{3} \\ 2 \, x_{1} + x_{2} - 6 \, x_{3} \\ -x_{2} + 6 \, x_{3} \end{array}\right]\]

\[\left[\begin{array}{r} 25 \\ 35 \\ 34 \\ -28 \end{array}\right]\]


Example 29

A2 - Linear maps and matrices (ver. 29)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} -x_{1} - 5 \, x_{2} - x_{3} + 5 \, x_{4} \\ x_{1} + 4 \, x_{2} + x_{3} - 4 \, x_{4} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 4 \\ 6 \\ -6 \\ 8 \end{array}\right] \right)\).

Answer.

\[\left[\begin{array}{rrrr} -1 & -5 & -1 & 5 \\ 1 & 4 & 1 & -4 \end{array}\right]\]

\[\left[\begin{array}{r} 12 \\ -10 \end{array}\right]\]


Example 30

A2 - Linear maps and matrices (ver. 30)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 6 \\ 5 \\ -4 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 4 \, x_{2} + 5 \, x_{3} \\ -x_{1} + 5 \, x_{2} + 7 \, x_{3} \\ x_{3} \\ -5 \, x_{3} \end{array}\right]\]

\[\left[\begin{array}{r} -34 \\ -9 \\ -4 \\ 20 \end{array}\right]\]


Example 31

A2 - Linear maps and matrices (ver. 31)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 5 \, x_{2} \\ x_{2} \\ 4 \, x_{2} \\ x_{1} - 6 \, x_{2} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -5 \\ -4 \end{array}\right] \right)\).

Answer.

\[\left[\begin{array}{rr} 1 & -5 \\ 0 & 1 \\ 0 & 4 \\ 1 & -6 \end{array}\right]\]

\[\left[\begin{array}{r} 15 \\ -4 \\ -16 \\ 19 \end{array}\right]\]


Example 32

A2 - Linear maps and matrices (ver. 32)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -2 \\ 3 \\ -3 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + x_{2} \\ -5 \, x_{1} - 4 \, x_{2} - 3 \, x_{3} \\ 3 \, x_{2} - 8 \, x_{3} \end{array}\right]\]

\[\left[\begin{array}{r} 1 \\ 7 \\ 33 \end{array}\right]\]


Example 33

A2 - Linear maps and matrices (ver. 33)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 4 \\ -6 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -26 \\ -20 \\ -12 \\ 10 \end{array}\right]\]


Example 34

A2 - Linear maps and matrices (ver. 34)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -6 \\ -6 \\ 3 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} \\ x_{2} + 4 \, x_{3} \\ x_{2} + 5 \, x_{3} \\ x_{1} - 2 \, x_{2} - 5 \, x_{3} \end{array}\right]\]

\[\left[\begin{array}{r} -6 \\ 6 \\ 9 \\ -9 \end{array}\right]\]


Example 35

A2 - Linear maps and matrices (ver. 35)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + 5 \, x_{2} \\ x_{1} + 6 \, x_{2} \\ x_{1} + 7 \, x_{2} \\ -2 \, x_{1} - 5 \, x_{2} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 6 \\ 6 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 36 \\ 42 \\ 48 \\ -42 \end{array}\right]\]


Example 36

A2 - Linear maps and matrices (ver. 36)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 1 \\ -2 \\ 0 \\ 8 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 3 \, x_{2} - 4 \, x_{3} - x_{4} \\ 2 \, x_{1} - 6 \, x_{2} - 7 \, x_{3} - 2 \, x_{4} \end{array}\right]\]

\[\left[\begin{array}{r} -1 \\ -2 \end{array}\right]\]


Example 37

A2 - Linear maps and matrices (ver. 37)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 6 \\ 2 \\ 5 \\ 4 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 10 \\ 5 \\ 2 \end{array}\right]\]


Example 38

A2 - Linear maps and matrices (ver. 38)

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

\[\left[\begin{array}{rr} -2 & -5 \\ -3 & -8 \\ 0 & -5 \end{array}\right]\]

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 6 \\ 0 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} -2 \, x_{1} - 5 \, x_{2} \\ -3 \, x_{1} - 8 \, x_{2} \\ -5 \, x_{2} \end{array}\right]\]

\[\left[\begin{array}{r} -12 \\ -18 \\ 0 \end{array}\right]\]


Example 39

A2 - Linear maps and matrices (ver. 39)

Let \(T:\mathbb R^{4}\to\mathbb R^{1}\) be the transformation given by

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

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 0 \\ 5 \\ 7 \\ -4 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 15 \end{array}\right]\]


Example 40

A2 - Linear maps and matrices (ver. 40)

Let \(T:\mathbb R^{3}\to\mathbb R^{3}\) be the transformation given by

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} -x_{1} + 4 \, x_{3} \\ -2 \, x_{1} - x_{2} + 8 \, x_{3} \\ 2 \, x_{1} - 3 \, x_{2} - 7 \, x_{3} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 2 \\ 8 \\ -5 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -22 \\ -52 \\ 15 \end{array}\right]\]


Example 41

A2 - Linear maps and matrices (ver. 41)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 3 \, x_{2} \\ 2 \, x_{1} - 5 \, x_{2} \\ 0 \\ 5 \, x_{2} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 2 \\ -1 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 5 \\ 9 \\ 0 \\ -5 \end{array}\right]\]


Example 42

A2 - Linear maps and matrices (ver. 42)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 6 \\ 0 \\ -4 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} -2 \, x_{1} - 2 \, x_{2} - x_{3} \\ x_{2} - x_{3} \\ -x_{1} + x_{2} - 3 \, x_{3} \\ x_{1} - 6 \, x_{2} + 8 \, x_{3} \end{array}\right]\]

\[\left[\begin{array}{r} -8 \\ 4 \\ 6 \\ -26 \end{array}\right]\]


Example 43

A2 - Linear maps and matrices (ver. 43)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 6 \\ -7 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 12 \\ 25 \\ 25 \\ -20 \end{array}\right]\]


Example 44

A2 - Linear maps and matrices (ver. 44)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -8 \\ 5 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 5 \, x_{2} \\ -x_{1} + 6 \, x_{2} \\ x_{2} \end{array}\right]\]

\[\left[\begin{array}{r} -33 \\ 38 \\ 5 \end{array}\right]\]


Example 45

A2 - Linear maps and matrices (ver. 45)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 3 \\ -1 \\ 4 \\ 0 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 5 \, x_{2} - 2 \, x_{4} \\ -x_{1} + 5 \, x_{2} + x_{3} + 3 \, x_{4} \end{array}\right]\]

\[\left[\begin{array}{r} 8 \\ -4 \end{array}\right]\]


Example 46

A2 - Linear maps and matrices (ver. 46)

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

\[\left[\begin{array}{rrr} -2 & -2 & -2 \\ -1 & 1 & 7 \\ 5 & 1 & -4 \\ 3 & 2 & 0 \end{array}\right]\]

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 7 \\ -4 \\ 8 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} -2 \, x_{1} - 2 \, x_{2} - 2 \, x_{3} \\ -x_{1} + x_{2} + 7 \, x_{3} \\ 5 \, x_{1} + x_{2} - 4 \, x_{3} \\ 3 \, x_{1} + 2 \, x_{2} \end{array}\right]\]

\[\left[\begin{array}{r} -22 \\ 45 \\ -1 \\ 13 \end{array}\right]\]


Example 47

A2 - Linear maps and matrices (ver. 47)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 6 \\ 7 \\ 2 \\ 7 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 24 \\ 44 \end{array}\right]\]


Example 48

A2 - Linear maps and matrices (ver. 48)

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

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

(a) Explain how to find an expression for \(T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)\)

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 4 \\ 8 \end{array}\right] \right)\).

Answer.

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \end{array}\right] \right)= \left[\begin{array}{r} -2 \, x_{1} - 5 \, x_{2} \\ -x_{1} - 3 \, x_{2} \\ -x_{1} - x_{2} \\ x_{2} \end{array}\right]\]

\[\left[\begin{array}{r} -48 \\ -28 \\ -12 \\ 8 \end{array}\right]\]


Example 49

A2 - Linear maps and matrices (ver. 49)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + 3 \, x_{2} + 2 \, x_{3} \\ -2 \, x_{1} - 5 \, x_{2} - 3 \, x_{3} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} -5 \\ -8 \\ -2 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} -33 \\ 56 \end{array}\right]\]


Example 50

A2 - Linear maps and matrices (ver. 50)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} -x_{2} + x_{3} \\ x_{1} - x_{2} \end{array}\right]\]

(a) Explain how to find the standard matrix for \(T\).

(b) Explain how to evaluate \(T\left( \left[\begin{array}{r} 8 \\ 0 \\ 3 \end{array}\right] \right)\).

Answer.

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

\[\left[\begin{array}{r} 3 \\ 8 \end{array}\right]\]