V8 - Dimension


Example 1

V8 - Dimension (ver. 1)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 2 \\ 2 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 1 \\ -2 \\ 2 \end{array}\right],\left[\begin{array}{r} -5 \\ -5 \\ -4 \\ -7 \\ 1 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 2

V8 - Dimension (ver. 2)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ 4 \\ 3 \\ 3 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ 8 \\ 3 \\ 6 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrr} 1 & 2 & 1 \\ 0 & 1 & 2 \\ 0 & 4 & 8 \\ 1 & 3 & 3 \\ 0 & 3 & 6 \end{array}\right]\sim\left[\begin{array}{rrr} 1 & 0 & -3 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(2\).


Example 3

V8 - Dimension (ver. 3)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 0 \\ -2 \end{array}\right],\left[\begin{array}{r} 6 \\ 2 \\ 0 \\ -4 \end{array}\right],\left[\begin{array}{r} 4 \\ 5 \\ 1 \\ -5 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} 1 & 1 & 6 & 4 \\ 0 & 1 & 2 & 5 \\ 0 & 0 & 0 & 1 \\ 0 & -2 & -4 & -5 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 4 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 4

V8 - Dimension (ver. 4)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} -3 \\ 1 \\ 2 \\ 3 \end{array}\right],\left[\begin{array}{r} 0 \\ -1 \\ 2 \\ 2 \end{array}\right],\left[\begin{array}{r} -6 \\ 4 \\ -1 \\ 0 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 5

V8 - Dimension (ver. 5)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} -3 \\ -3 \\ -2 \end{array}\right],\left[\begin{array}{r} -3 \\ -4 \\ -3 \end{array}\right],\left[\begin{array}{r} -7 \\ -5 \\ -5 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 6

V8 - Dimension (ver. 6)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 2 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 2 \\ 6 \\ 1 \end{array}\right],\left[\begin{array}{r} 0 \\ 2 \\ 5 \\ 7 \\ 4 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrr} 1 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 2 & 5 \\ 2 & 6 & 7 \\ 0 & 1 & 4 \end{array}\right]\sim\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 7

V8 - Dimension (ver. 7)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ -1 \\ 1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ -2 \\ -1 \\ -2 \end{array}\right],\left[\begin{array}{r} -2 \\ -5 \\ 0 \\ 3 \\ 2 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 8

V8 - Dimension (ver. 8)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 5 \\ -4 \\ -5 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 4 \\ -3 \\ -4 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 1 \\ 6 \\ 1 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrr} 5 & 4 & 0 \\ -4 & -3 & 1 \\ -5 & -4 & 1 \\ 1 & 1 & 6 \\ 0 & 0 & 1 \end{array}\right]\sim\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 9

V8 - Dimension (ver. 9)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 0 \\ 1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 0 \\ -3 \\ -3 \\ -3 \end{array}\right],\left[\begin{array}{r} 0 \\ 3 \\ 2 \\ 2 \end{array}\right],\left[\begin{array}{r} 0 \\ 8 \\ 6 \\ 6 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(2\).


Example 10

V8 - Dimension (ver. 10)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 7 \\ 4 \\ 2 \\ 3 \\ 4 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 2 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} 7 \\ 1 \\ -4 \\ 3 \\ 7 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(2\).


Example 11

V8 - Dimension (ver. 11)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 2 \\ -1 \\ 0 \\ -4 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ -3 \end{array}\right],\left[\begin{array}{r} -2 \\ 2 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{r} 0 \\ -4 \\ -3 \\ 8 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 12

V8 - Dimension (ver. 12)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ -4 \\ 0 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ 5 \\ -2 \\ 2 \end{array}\right],\left[\begin{array}{r} -1 \\ 8 \\ -7 \\ 4 \end{array}\right],\left[\begin{array}{r} -3 \\ 8 \\ 6 \\ 3 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} 1 & -1 & -1 & -3 \\ -4 & 5 & 8 & 8 \\ 0 & -2 & -7 & 6 \\ -2 & 2 & 4 & 3 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The dimension of \(W\) is \(4\).


Example 13

V8 - Dimension (ver. 13)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 0 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{r} -1 \\ -2 \\ 3 \end{array}\right],\left[\begin{array}{r} -4 \\ -7 \\ 3 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 14

V8 - Dimension (ver. 14)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ -1 \\ -2 \end{array}\right],\left[\begin{array}{r} 4 \\ 3 \\ 2 \end{array}\right],\left[\begin{array}{r} 7 \\ 6 \\ 8 \end{array}\right],\left[\begin{array}{r} 7 \\ 5 \\ 2 \end{array}\right],\left[\begin{array}{r} 4 \\ 3 \\ 2 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrrr} -1 & 4 & 7 & 7 & 4 \\ -1 & 3 & 6 & 5 & 3 \\ -2 & 2 & 8 & 2 & 2 \end{array}\right]\sim\left[\begin{array}{rrrrr} 1 & 0 & -3 & 1 & 0 \\ 0 & 1 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(2\).


Example 15

V8 - Dimension (ver. 15)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ -1 \\ 1 \\ -1 \\ -2 \end{array}\right],\left[\begin{array}{r} 5 \\ 4 \\ -1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 8 \\ 6 \\ 1 \\ -3 \\ 0 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrr} -1 & 5 & 8 \\ -1 & 4 & 6 \\ 1 & -1 & 1 \\ -1 & 1 & -3 \\ -2 & 1 & 0 \end{array}\right]\sim\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 16

V8 - Dimension (ver. 16)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 0 \\ 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ 3 \\ -4 \\ 3 \end{array}\right],\left[\begin{array}{r} -1 \\ 3 \\ -3 \\ 2 \end{array}\right],\left[\begin{array}{r} -3 \\ -8 \\ 4 \\ 3 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(4\).


Example 17

V8 - Dimension (ver. 17)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} 0 \\ 4 \\ 5 \\ -3 \end{array}\right],\left[\begin{array}{r} -1 \\ 6 \\ 8 \\ -5 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} 1 & 1 & 0 & -1 \\ 0 & 1 & 4 & 6 \\ 0 & 1 & 5 & 8 \\ -1 & -1 & -3 & -5 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 18

V8 - Dimension (ver. 18)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 4 \\ -1 \\ 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} -4 \\ -1 \\ -2 \\ -6 \\ 4 \end{array}\right],\left[\begin{array}{r} 8 \\ 0 \\ 3 \\ 5 \\ -4 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrr} 4 & -4 & 8 \\ -1 & -1 & 0 \\ 1 & -2 & 3 \\ -1 & -6 & 5 \\ 0 & 4 & -4 \end{array}\right]\sim\left[\begin{array}{rrr} 1 & 0 & 1 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(2\).


Example 19

V8 - Dimension (ver. 19)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ -2 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 4 \\ -7 \\ 3 \\ 7 \end{array}\right],\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ 4 \end{array}\right],\left[\begin{array}{r} 5 \\ -8 \\ 4 \\ 3 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} 1 & 4 & 0 & 5 \\ -2 & -7 & 0 & -8 \\ 0 & 3 & 1 & 4 \\ 1 & 7 & 4 & 3 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 20

V8 - Dimension (ver. 20)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 3 \\ 0 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} 1 \\ 4 \\ -2 \\ -3 \end{array}\right],\left[\begin{array}{r} -1 \\ 5 \\ -3 \\ -1 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 21

V8 - Dimension (ver. 21)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} -1 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} -6 \\ -1 \\ 4 \end{array}\right],\left[\begin{array}{r} -2 \\ 0 \\ 2 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(2\).


Example 22

V8 - Dimension (ver. 22)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ -1 \\ 2 \end{array}\right],\left[\begin{array}{r} -1 \\ -2 \\ 3 \end{array}\right],\left[\begin{array}{r} -2 \\ -3 \\ 5 \end{array}\right],\left[\begin{array}{r} -3 \\ -5 \\ 8 \end{array}\right],\left[\begin{array}{r} 2 \\ 3 \\ -5 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrrr} -1 & -1 & -2 & -3 & 2 \\ -1 & -2 & -3 & -5 & 3 \\ 2 & 3 & 5 & 8 & -5 \end{array}\right]\sim\left[\begin{array}{rrrrr} 1 & 0 & 1 & 1 & -1 \\ 0 & 1 & 1 & 2 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(2\).


Example 23

V8 - Dimension (ver. 23)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ 2 \end{array}\right],\left[\begin{array}{r} -2 \\ -7 \\ -6 \end{array}\right],\left[\begin{array}{r} -3 \\ -6 \\ -6 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrrr} 1 & 2 & 1 & -2 & -3 \\ -1 & -2 & 2 & -7 & -6 \\ 0 & 0 & 2 & -6 & -6 \end{array}\right]\sim\left[\begin{array}{rrrrr} 1 & 2 & 0 & 1 & 0 \\ 0 & 0 & 1 & -3 & -3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(2\).


Example 24

V8 - Dimension (ver. 24)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 3 \\ -2 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 8 \\ -5 \\ 3 \\ -2 \end{array}\right],\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ 3 \end{array}\right],\left[\begin{array}{r} 4 \\ -3 \\ -6 \\ -2 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} 3 & 8 & 0 & 4 \\ -2 & -5 & 0 & -3 \\ 0 & 3 & 1 & -6 \\ 1 & -2 & 3 & -2 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\]

The dimension of \(W\) is \(4\).


Example 25

V8 - Dimension (ver. 25)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ 2 \\ -1 \\ 1 \end{array}\right],\left[\begin{array}{r} 0 \\ -1 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} 0 \\ 4 \\ -3 \\ 7 \end{array}\right],\left[\begin{array}{r} -3 \\ 0 \\ 1 \\ -7 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 26

V8 - Dimension (ver. 26)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 1 \\ 0 \\ -1 \\ 1 \end{array}\right],\left[\begin{array}{r} 3 \\ 5 \\ -1 \\ -4 \\ 1 \end{array}\right],\left[\begin{array}{r} 7 \\ 8 \\ 0 \\ -2 \\ 8 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrr} 1 & 3 & 7 \\ 1 & 5 & 8 \\ 0 & -1 & 0 \\ -1 & -4 & -2 \\ 1 & 1 & 8 \end{array}\right]\sim\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 27

V8 - Dimension (ver. 27)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} -4 \\ 3 \\ -3 \end{array}\right],\left[\begin{array}{r} 6 \\ -5 \\ 4 \end{array}\right],\left[\begin{array}{r} 2 \\ -1 \\ 2 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 28

V8 - Dimension (ver. 28)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ -1 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{r} 0 \\ 0 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ -3 \\ -6 \end{array}\right],\left[\begin{array}{r} 2 \\ -1 \\ -3 \\ -8 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} -1 & 0 & 2 & 2 \\ -1 & 0 & 1 & -1 \\ 1 & 1 & -3 & -3 \\ 2 & 1 & -6 & -8 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 29

V8 - Dimension (ver. 29)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 2 \\ 1 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} 1 \\ 3 \\ -3 \\ 4 \\ -1 \end{array}\right],\left[\begin{array}{r} -4 \\ -7 \\ -7 \\ 0 \\ 6 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrr} 1 & 1 & -4 \\ 2 & 3 & -7 \\ 1 & -3 & -7 \\ 1 & 4 & 0 \\ -2 & -1 & 6 \end{array}\right]\sim\left[\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 30

V8 - Dimension (ver. 30)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ 2 \\ -2 \end{array}\right],\left[\begin{array}{r} 6 \\ 0 \\ 1 \\ -3 \end{array}\right],\left[\begin{array}{r} 3 \\ -5 \\ -7 \\ 2 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(4\).


Example 31

V8 - Dimension (ver. 31)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} 4 \\ 0 \\ -3 \end{array}\right],\left[\begin{array}{r} 8 \\ -5 \\ -2 \end{array}\right],\left[\begin{array}{r} -3 \\ -1 \\ 3 \end{array}\right],\left[\begin{array}{r} 6 \\ -8 \\ 2 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrrr} 1 & 4 & 8 & -3 & 6 \\ -1 & 0 & -5 & -1 & -8 \\ 0 & -3 & -2 & 3 & 2 \end{array}\right]\sim\left[\begin{array}{rrrrr} 1 & 0 & 0 & 1 & -2 \\ 0 & 1 & 0 & -1 & -2 \\ 0 & 0 & 1 & 0 & 2 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 32

V8 - Dimension (ver. 32)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ -1 \\ -1 \\ 1 \end{array}\right],\left[\begin{array}{r} -3 \\ 2 \\ -1 \\ 1 \end{array}\right],\left[\begin{array}{r} 3 \\ 5 \\ 6 \\ -4 \end{array}\right],\left[\begin{array}{r} -8 \\ 2 \\ -4 \\ 4 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} -1 & -3 & 3 & -8 \\ -1 & 2 & 5 & 2 \\ -1 & -1 & 6 & -4 \\ 1 & 1 & -4 & 4 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 33

V8 - Dimension (ver. 33)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ 5 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} -4 \\ -2 \\ -2 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(2\).


Example 34

V8 - Dimension (ver. 34)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ -2 \\ 1 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ -6 \\ 1 \end{array}\right],\left[\begin{array}{r} -2 \\ 3 \\ -6 \\ 0 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(4\).


Example 35

V8 - Dimension (ver. 35)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -4 \\ -3 \\ 0 \end{array}\right],\left[\begin{array}{r} 4 \\ 4 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ 3 \\ 2 \end{array}\right],\left[\begin{array}{r} -8 \\ -6 \\ 0 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrrr} 1 & -4 & 4 & 1 & -8 \\ 1 & -3 & 4 & 3 & -6 \\ 0 & 0 & 1 & 2 & 0 \end{array}\right]\sim\left[\begin{array}{rrrrr} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 2 & 2 \\ 0 & 0 & 1 & 2 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 36

V8 - Dimension (ver. 36)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 2 \\ 2 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ -1 \\ 2 \\ -3 \\ 2 \end{array}\right],\left[\begin{array}{r} -3 \\ 0 \\ -6 \\ 7 \\ -4 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrr} 1 & 1 & -3 \\ 2 & -1 & 0 \\ 2 & 2 & -6 \\ -1 & -3 & 7 \\ 0 & 2 & -4 \end{array}\right]\sim\left[\begin{array}{rrr} 1 & 0 & -1 \\ 0 & 1 & -2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(2\).


Example 37

V8 - Dimension (ver. 37)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 2 \\ -1 \\ 2 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ -1 \\ 2 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 2 \\ -2 \\ -6 \\ -3 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 38

V8 - Dimension (ver. 38)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ 0 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrrr} 1 & 1 & 1 & -2 & -1 \\ 0 & 1 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\sim\left[\begin{array}{rrrrr} 1 & 0 & -1 & -1 & -2 \\ 0 & 1 & 2 & -1 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(2\).


Example 39

V8 - Dimension (ver. 39)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ -1 \\ -1 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} 5 \\ -4 \\ -7 \\ 0 \\ -5 \end{array}\right],\left[\begin{array}{r} -2 \\ 2 \\ 2 \\ -2 \\ 2 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(2\).


Example 40

V8 - Dimension (ver. 40)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 3 \\ 2 \\ 0 \\ 3 \end{array}\right],\left[\begin{array}{r} -5 \\ -6 \\ -3 \\ -8 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} 1 & 0 & 3 & -5 \\ 0 & 1 & 2 & -6 \\ -1 & 1 & 0 & -3 \\ 0 & 1 & 3 & -8 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 41

V8 - Dimension (ver. 41)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{r} 0 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ -3 \\ -8 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 2 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 42

V8 - Dimension (ver. 42)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ 1 \\ 2 \end{array}\right],\left[\begin{array}{r} -2 \\ 0 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} 0 \\ -2 \\ -4 \\ 0 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 43

V8 - Dimension (ver. 43)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right],\left[\begin{array}{r} 4 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} -3 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{r} 7 \\ 0 \\ -6 \end{array}\right],\left[\begin{array}{r} -7 \\ 0 \\ 6 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrrr} 1 & 4 & -3 & 7 & -7 \\ 0 & 1 & -2 & 0 & 0 \\ -1 & -2 & 0 & -6 & 6 \end{array}\right]\sim\left[\begin{array}{rrrrr} 1 & 0 & 0 & 2 & -2 \\ 0 & 1 & 0 & 2 & -2 \\ 0 & 0 & 1 & 1 & -1 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 44

V8 - Dimension (ver. 44)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ 1 \end{array}\right],\left[\begin{array}{r} 3 \\ -5 \\ -1 \end{array}\right],\left[\begin{array}{r} 3 \\ -3 \\ -3 \end{array}\right],\left[\begin{array}{r} 2 \\ -4 \\ 0 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(2\).


Example 45

V8 - Dimension (ver. 45)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ -2 \\ 2 \\ -1 \\ -3 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 2 \\ -2 \\ -2 \end{array}\right],\left[\begin{array}{r} 2 \\ -8 \\ -3 \\ 4 \\ -1 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(3\).


Example 46

V8 - Dimension (ver. 46)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 2 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} -3 \\ 1 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 8 \\ 2 \\ -1 \\ 1 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(4\).


Example 47

V8 - Dimension (ver. 47)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 1 \\ 1 \\ 0 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} 0 \\ -3 \\ 4 \\ -5 \\ 2 \end{array}\right],\left[\begin{array}{r} -2 \\ 4 \\ -8 \\ 8 \\ -2 \end{array}\right]\right\}.\]

Answer.

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

The dimension of \(W\) is \(2\).


Example 48

V8 - Dimension (ver. 48)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} -1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} -3 \\ 1 \\ 3 \end{array}\right],\left[\begin{array}{r} -3 \\ -1 \\ 2 \end{array}\right],\left[\begin{array}{r} -3 \\ 5 \\ 5 \end{array}\right],\left[\begin{array}{r} 8 \\ -2 \\ -8 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrrr} -1 & -3 & -3 & -3 & 8 \\ 0 & 1 & -1 & 5 & -2 \\ 1 & 3 & 2 & 5 & -8 \end{array}\right]\sim\left[\begin{array}{rrrrr} 1 & 0 & 0 & 0 & -2 \\ 0 & 1 & 0 & 3 & -2 \\ 0 & 0 & 1 & -2 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 49

V8 - Dimension (ver. 49)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 0 \\ 1 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} -1 \\ -4 \\ 4 \\ 1 \end{array}\right],\left[\begin{array}{r} 0 \\ 4 \\ -3 \\ -7 \end{array}\right],\left[\begin{array}{r} -1 \\ -7 \\ 6 \\ 7 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} 0 & -1 & 0 & -1 \\ 1 & -4 & 4 & -7 \\ -1 & 4 & -3 & 6 \\ -1 & 1 & -7 & 7 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(3\).


Example 50

V8 - Dimension (ver. 50)

Explain how to find the dimension of

\[W=\mathrm{span}\left\{\left[\begin{array}{r} 5 \\ -1 \\ 3 \\ -4 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{r} -7 \\ -1 \\ -7 \\ 6 \end{array}\right],\left[\begin{array}{r} 4 \\ -2 \\ 1 \\ -3 \end{array}\right]\right\}.\]

Answer.

\[\left[\begin{array}{rrrr} 5 & 1 & -7 & 4 \\ -1 & 1 & -1 & -2 \\ 3 & 2 & -7 & 1 \\ -4 & -1 & 6 & -3 \end{array}\right]\sim\left[\begin{array}{rrrr} 1 & 0 & -1 & 1 \\ 0 & 1 & -2 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]\]

The dimension of \(W\) is \(2\).