A3 - Kernel and Image


Example 1

A3 - Kernel and Image (ver. 1)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + x_{2} - 6 \, x_{3} + 8 \, x_{4} \\ x_{2} - 5 \, x_{3} + 6 \, x_{4} \\ -2 \, x_{1} + x_{2} + 4 \, x_{3} - 5 \, x_{4} \\ -x_{2} + 2 \, x_{3} - 3 \, x_{4} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} -6 \\ -5 \\ 4 \\ 2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ -1 \\ 1 \\ 1 \end{array}\right]\right\}\)


Example 2

A3 - Kernel and Image (ver. 2)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} 0 \\ 2 \, x_{1} + 7 \, x_{2} + 8 \, x_{3} - 7 \, x_{4} \\ -2 \, x_{1} - 6 \, x_{2} - 6 \, x_{3} + 6 \, x_{4} \\ -x_{1} - 3 \, x_{2} - 3 \, x_{3} + 3 \, x_{4} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 0 \\ 2 \\ -2 \\ -1 \end{array}\right],\left[\begin{array}{r} 0 \\ 7 \\ -6 \\ -3 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 3 \\ -2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 3

A3 - Kernel and Image (ver. 3)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -2 \\ 3 \\ 1 \\ 3 \end{array}\right],\left[\begin{array}{r} 1 \\ -8 \\ 2 \\ -4 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -2 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 4

A3 - Kernel and Image (ver. 4)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 0 \\ 1 \\ 2 \\ -3 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 0 \\ 7 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 5

A3 - Kernel and Image (ver. 5)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -1 \\ 2 \\ 0 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ -1 \\ -2 \\ 0 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 3 \\ 1 \\ 0 \end{array}\right]\right\}\)


Example 6

A3 - Kernel and Image (ver. 6)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ -5 \\ 0 \\ 3 \end{array}\right],\left[\begin{array}{r} 1 \\ -4 \\ 0 \\ 4 \end{array}\right],\left[\begin{array}{r} -3 \\ 0 \\ 1 \\ -6 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 0 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 7

A3 - Kernel and Image (ver. 7)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 2 \\ -2 \\ 1 \end{array}\right],\left[\begin{array}{r} -1 \\ -1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -3 \\ -2 \\ 3 \\ -2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ -3 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 8

A3 - Kernel and Image (ver. 8)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ -3 \\ -2 \\ -3 \end{array}\right],\left[\begin{array}{r} 1 \\ -2 \\ -2 \\ -1 \end{array}\right],\left[\begin{array}{r} -3 \\ 5 \\ 7 \\ 4 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -3 \\ -1 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 9

A3 - Kernel and Image (ver. 9)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 5 \\ 3 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 5 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 10

A3 - Kernel and Image (ver. 10)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -7 \\ 3 \\ 2 \\ 4 \end{array}\right],\left[\begin{array}{r} -4 \\ 1 \\ -1 \\ 2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 3 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ -2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 11

A3 - Kernel and Image (ver. 11)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -1 \\ 0 \\ 2 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 1 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ 2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 12

A3 - Kernel and Image (ver. 12)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{r} 8 \\ -3 \\ -5 \\ 4 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 1 \\ 1 \end{array}\right]\right\}\)


Example 13

A3 - Kernel and Image (ver. 13)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -7 \\ -4 \\ 2 \\ -4 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 1 \\ 0 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 0 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 14

A3 - Kernel and Image (ver. 14)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} -3 \\ 4 \\ 4 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -4 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 15

A3 - Kernel and Image (ver. 15)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + x_{2} - 3 \, x_{3} - 3 \, x_{4} \\ 2 \, x_{1} - x_{2} - 8 \, x_{3} + 6 \, x_{4} \\ -x_{2} - x_{3} + 4 \, x_{4} \\ x_{1} - 5 \, x_{3} + x_{4} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 2 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ -1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} -3 \\ -8 \\ -1 \\ -5 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 4 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 16

A3 - Kernel and Image (ver. 16)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 0 \\ 0 \\ -1 \\ 1 \end{array}\right],\left[\begin{array}{r} -4 \\ 1 \\ 1 \\ -4 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -4 \\ 1 \\ 0 \end{array}\right]\right\}\)


Example 17

A3 - Kernel and Image (ver. 17)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ 2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 18

A3 - Kernel and Image (ver. 18)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} -x_{1} - 6 \, x_{2} + 5 \, x_{3} + 8 \, x_{4} \\ -x_{1} - 3 \, x_{2} + 2 \, x_{3} + 5 \, x_{4} \\ -x_{2} + x_{3} + x_{4} \\ x_{1} + 5 \, x_{2} - 4 \, x_{3} - 7 \, x_{4} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -1 \\ -1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} -6 \\ -3 \\ -1 \\ 5 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 19

A3 - Kernel and Image (ver. 19)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} -x_{1} + x_{2} + 5 \, x_{3} \\ 2 \, x_{1} - x_{2} - 8 \, x_{3} \\ 0 \\ x_{1} - x_{2} - 5 \, x_{3} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -1 \\ 2 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ -1 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 3 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 20

A3 - Kernel and Image (ver. 20)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} + x_{2} - 5 \, x_{3} \\ x_{1} + 2 \, x_{2} - 8 \, x_{3} \\ -2 \, x_{1} + 4 \, x_{3} \\ -2 \, x_{2} + 6 \, x_{3} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 1 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ 0 \\ -2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 21

A3 - Kernel and Image (ver. 21)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} -2 \, x_{1} + 2 \, x_{2} + 4 \, x_{3} - 4 \, x_{4} \\ -4 \, x_{1} + x_{2} + 8 \, x_{3} + x_{4} \\ 3 \, x_{1} - 2 \, x_{2} - 6 \, x_{3} + 3 \, x_{4} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -2 \\ -4 \\ 3 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ -2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 3 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 22

A3 - Kernel and Image (ver. 22)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 0 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ -2 \\ 4 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -2 \\ -3 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ -3 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 23

A3 - Kernel and Image (ver. 23)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ -3 \end{array}\right],\left[\begin{array}{r} -2 \\ 3 \\ 3 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ -1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 24

A3 - Kernel and Image (ver. 24)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -8 \\ 3 \\ 3 \\ -5 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 0 \\ 3 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 0 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 25

A3 - Kernel and Image (ver. 25)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -1 \\ 0 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} -5 \\ 1 \\ 0 \\ -1 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -5 \\ 1 \\ 0 \end{array}\right]\right\}\)


Example 26

A3 - Kernel and Image (ver. 26)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ -1 \\ 6 \end{array}\right],\left[\begin{array}{r} -3 \\ 0 \\ 0 \\ -2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ -1 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 27

A3 - Kernel and Image (ver. 27)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ -2 \\ 2 \\ -1 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -2 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 28

A3 - Kernel and Image (ver. 28)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -1 \\ -1 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ -1 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -4 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 29

A3 - Kernel and Image (ver. 29)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -5 \\ 1 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ 2 \\ 0 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 0 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 30

A3 - Kernel and Image (ver. 30)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 3 \, x_{2} - 5 \, x_{3} \\ x_{2} + x_{3} \\ -x_{1} + 4 \, x_{2} + 6 \, x_{3} \\ -2 \, x_{1} + 4 \, x_{2} + 8 \, x_{3} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ -1 \\ -2 \end{array}\right],\left[\begin{array}{r} -3 \\ 1 \\ 4 \\ 4 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 31

A3 - Kernel and Image (ver. 31)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 5 \\ 5 \\ 1 \end{array}\right],\left[\begin{array}{r} -1 \\ -4 \\ -5 \\ -6 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ 1 \\ 1 \end{array}\right]\right\}\)


Example 32

A3 - Kernel and Image (ver. 32)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 4 \\ -1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 7 \\ -3 \\ 4 \\ 3 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 3 \\ 0 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 33

A3 - Kernel and Image (ver. 33)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 2 \, x_{2} - x_{3} + 5 \, x_{4} \\ x_{1} - x_{2} + 3 \, x_{4} \\ 0 \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ -1 \\ 0 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ -1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 34

A3 - Kernel and Image (ver. 34)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 5 \, x_{2} - 2 \, x_{3} - 8 \, x_{4} \\ -x_{1} + 6 \, x_{2} + 3 \, x_{3} + 8 \, x_{4} \\ -x_{1} + 5 \, x_{2} + 3 \, x_{3} + 5 \, x_{4} \\ -5 \, x_{2} - 6 \, x_{3} + 3 \, x_{4} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} -5 \\ 6 \\ 5 \\ -5 \end{array}\right],\left[\begin{array}{r} -2 \\ 3 \\ 3 \\ -6 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ -3 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 35

A3 - Kernel and Image (ver. 35)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 1 \\ -1 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -3 \\ -3 \\ 1 \end{array}\right]\right\}\)


Example 36

A3 - Kernel and Image (ver. 36)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 0 \\ 2 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} -2 \\ -7 \\ -2 \\ -2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 3 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 37

A3 - Kernel and Image (ver. 37)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 5 \\ -2 \\ -4 \\ 1 \end{array}\right],\left[\begin{array}{r} 3 \\ -1 \\ -3 \\ 1 \end{array}\right],\left[\begin{array}{r} -4 \\ 1 \\ 6 \\ -2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ -1 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 38

A3 - Kernel and Image (ver. 38)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ 5 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -3 \\ 2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 39

A3 - Kernel and Image (ver. 39)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} 3 \, x_{1} - 8 \, x_{2} - 6 \, x_{3} - 4 \, x_{4} \\ -2 \, x_{1} + x_{2} + 4 \, x_{3} + 7 \, x_{4} \\ -2 \, x_{1} + 3 \, x_{2} + 4 \, x_{3} + 5 \, x_{4} \\ -x_{1} + 4 \, x_{2} + 2 \, x_{3} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 3 \\ -2 \\ -2 \\ -1 \end{array}\right],\left[\begin{array}{r} -8 \\ 1 \\ 3 \\ 4 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 4 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 40

A3 - Kernel and Image (ver. 40)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ 2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ 2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -3 \\ -3 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 41

A3 - Kernel and Image (ver. 41)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} 6 \\ 1 \\ 5 \\ -5 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -1 \\ -1 \\ 1 \end{array}\right]\right\}\)


Example 42

A3 - Kernel and Image (ver. 42)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \end{array}\right] \right)= \left[\begin{array}{r} x_{1} - 4 \, x_{2} + 8 \, x_{3} \\ x_{1} - 3 \, x_{2} + 6 \, x_{3} \\ -x_{1} + 3 \, x_{2} - 6 \, x_{3} \\ 0 \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} -4 \\ -3 \\ 3 \\ 0 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 0 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 43

A3 - Kernel and Image (ver. 43)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -2 \\ 3 \\ 0 \\ 2 \end{array}\right],\left[\begin{array}{r} 1 \\ -2 \\ -2 \\ 0 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -2 \\ -3 \\ 1 \end{array}\right]\right\}\)


Example 44

A3 - Kernel and Image (ver. 44)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} 0 \\ 3 \, x_{1} + x_{2} - 3 \, x_{3} - 7 \, x_{4} \\ x_{1} - x_{3} - 3 \, x_{4} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 0 \\ 3 \\ 1 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 0 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ -2 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 45

A3 - Kernel and Image (ver. 45)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} 2 \\ -1 \\ 2 \\ 0 \end{array}\right],\left[\begin{array}{r} -4 \\ 2 \\ -3 \\ 2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ 1 \\ 3 \\ 1 \end{array}\right]\right\}\)


Example 46

A3 - Kernel and Image (ver. 46)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -1 \\ 1 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{r} 4 \\ 5 \\ 8 \\ 0 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 47

A3 - Kernel and Image (ver. 47)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} -2 \\ 0 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ 0 \\ -3 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 2 \\ 2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\)


Example 48

A3 - Kernel and Image (ver. 48)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ -3 \\ 2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -2 \\ 2 \\ 1 \end{array}\right]\right\}\)


Example 49

A3 - Kernel and Image (ver. 49)

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

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

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 0 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ -3 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} -4 \\ 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ 0 \\ -2 \\ 1 \end{array}\right]\right\}\)


Example 50

A3 - Kernel and Image (ver. 50)

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

\[T\left( \left[\begin{array}{r} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right] \right)= \left[\begin{array}{r} 4 \, x_{1} - 8 \, x_{2} + 4 \, x_{3} \\ 2 \, x_{1} - 3 \, x_{2} + x_{3} + x_{4} \\ -x_{1} + 7 \, x_{2} - 6 \, x_{3} + 5 \, x_{4} \\ -x_{1} + 2 \, x_{2} - x_{3} \end{array}\right].\]

Show how to compute a basis for its image, and a basis for its kernel.

Answer.

A basis for the image: \(\left\{\left[\begin{array}{r} 4 \\ 2 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} -8 \\ -3 \\ 7 \\ 2 \end{array}\right]\right\}\)

A basis for the kernel: \(\left\{\left[\begin{array}{r} 1 \\ 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ -1 \\ 0 \\ 1 \end{array}\right]\right\}\)