V7 - Basis Computation


Example 1

V7 - Basis Computation (ver. 1)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 2 \\ 0 \\ 1 \\ 2 \\ 2 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ -1 \\ 2 \\ 3 \end{array}\right]\right\}\).


Example 2

V7 - Basis Computation (ver. 2)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ -4 \\ 3 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ 5 \\ -4 \\ 4 \end{array}\right]\right\}\).


Example 3

V7 - Basis Computation (ver. 3)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 1 \\ -2 \\ -2 \end{array}\right],\left[\begin{array}{r} 5 \\ 6 \\ -5 \\ -7 \end{array}\right]\right\}\).


Example 4

V7 - Basis Computation (ver. 4)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 2 \\ -3 \end{array}\right],\left[\begin{array}{r} -1 \\ 3 \\ -5 \\ 8 \end{array}\right]\right\}\).


Example 5

V7 - Basis Computation (ver. 5)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 3 \\ 4 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ -2 \end{array}\right]\right\}\).


Example 6

V7 - Basis Computation (ver. 6)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 2 \\ -2 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} -6 \\ -1 \\ 4 \\ 3 \\ -5 \end{array}\right]\right\}\).


Example 7

V7 - Basis Computation (ver. 7)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -5 \\ -3 \\ -2 \\ -3 \\ 4 \end{array}\right],\left[\begin{array}{r} 4 \\ -2 \\ 1 \\ -2 \\ 8 \end{array}\right]\right\}\).


Example 8

V7 - Basis Computation (ver. 8)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -3 \\ -2 \\ 1 \\ 0 \\ 2 \end{array}\right],\left[\begin{array}{r} 1 \\ -1 \\ 2 \\ 2 \\ 0 \end{array}\right]\right\}\).


Example 9

V7 - Basis Computation (ver. 9)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \\ -2 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} 4 \\ 3 \\ 1 \\ -5 \end{array}\right]\right\}\).


Example 10

V7 - Basis Computation (ver. 10)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -7 \\ 4 \\ 2 \\ 3 \\ 2 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ 2 \\ 0 \\ -1 \end{array}\right]\right\}\).


Example 11

V7 - Basis Computation (ver. 11)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -1 \\ -1 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{r} 0 \\ -1 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} 1 \\ 3 \\ -3 \\ 0 \end{array}\right]\right\}\).


Example 12

V7 - Basis Computation (ver. 12)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ -3 \\ 2 \\ 3 \end{array}\right]\right\}\).


Example 13

V7 - Basis Computation (ver. 13)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} -4 \\ 1 \\ 2 \end{array}\right]\right\}\).


Example 14

V7 - Basis Computation (ver. 14)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ -2 \\ -2 \end{array}\right]\right\}\).


Example 15

V7 - Basis Computation (ver. 15)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -2 \\ 3 \\ -2 \\ -2 \end{array}\right],\left[\begin{array}{r} -3 \\ 7 \\ -7 \\ 3 \\ 8 \end{array}\right]\right\}\).


Example 16

V7 - Basis Computation (ver. 16)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 3 \\ 2 \\ 3 \\ -3 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ 1 \\ -1 \end{array}\right]\right\}\).


Example 17

V7 - Basis Computation (ver. 17)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} -6 \\ -5 \\ 2 \\ -3 \end{array}\right]\right\}\).


Example 18

V7 - Basis Computation (ver. 18)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -3 \\ -1 \\ 0 \\ -2 \\ 2 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ 2 \\ -1 \\ 1 \end{array}\right]\right\}\).


Example 19

V7 - Basis Computation (ver. 19)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -1 \\ 4 \\ -4 \\ -2 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ 2 \\ 2 \end{array}\right]\right\}\).


Example 20

V7 - Basis Computation (ver. 20)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 1 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 2 \\ 0 \\ -2 \end{array}\right]\right\}\).


Example 21

V7 - Basis Computation (ver. 21)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -2 \\ 0 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ -2 \end{array}\right]\right\}\).


Example 22

V7 - Basis Computation (ver. 22)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -1 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{r} 1 \\ -3 \\ 5 \end{array}\right]\right\}\).


Example 23

V7 - Basis Computation (ver. 23)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 2 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} -1 \\ 3 \\ -1 \end{array}\right]\right\}\).


Example 24

V7 - Basis Computation (ver. 24)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 3 \\ -2 \\ 5 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 3 \\ 3 \end{array}\right]\right\}\).


Example 25

V7 - Basis Computation (ver. 25)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -1 \\ 2 \\ -1 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ -4 \\ -1 \end{array}\right]\right\}\).


Example 26

V7 - Basis Computation (ver. 26)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 3 \\ -4 \\ 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} -3 \\ 5 \\ -1 \\ -3 \\ 3 \end{array}\right]\right\}\).


Example 27

V7 - Basis Computation (ver. 27)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 2 \\ 3 \end{array}\right],\left[\begin{array}{r} -1 \\ -1 \\ 0 \end{array}\right]\right\}\).


Example 28

V7 - Basis Computation (ver. 28)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -2 \\ -1 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} 5 \\ 2 \\ -6 \\ 7 \end{array}\right]\right\}\).


Example 29

V7 - Basis Computation (ver. 29)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 4 \\ 3 \\ 1 \\ 4 \\ -4 \end{array}\right],\left[\begin{array}{r} 5 \\ 4 \\ -2 \\ 5 \\ -8 \end{array}\right]\right\}\).


Example 30

V7 - Basis Computation (ver. 30)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ -2 \\ -2 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ -4 \\ -3 \end{array}\right]\right\}\).


Example 31

V7 - Basis Computation (ver. 31)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 2 \\ -1 \\ 1 \end{array}\right],\left[\begin{array}{r} 1 \\ 0 \\ 2 \end{array}\right]\right\}\).


Example 32

V7 - Basis Computation (ver. 32)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -1 \\ -2 \\ 2 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ 1 \\ -1 \end{array}\right]\right\}\).


Example 33

V7 - Basis Computation (ver. 33)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -4 \\ 0 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ 1 \end{array}\right]\right\}\).


Example 34

V7 - Basis Computation (ver. 34)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ 2 \\ 0 \\ 1 \end{array}\right]\right\}\).


Example 35

V7 - Basis Computation (ver. 35)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ 2 \end{array}\right]\right\}\).


Example 36

V7 - Basis Computation (ver. 36)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -1 \\ 1 \\ 1 \\ 1 \\ -1 \end{array}\right],\left[\begin{array}{r} -1 \\ 0 \\ 1 \\ 1 \\ 3 \end{array}\right]\right\}\).


Example 37

V7 - Basis Computation (ver. 37)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -2 \\ -1 \\ 2 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{r} 3 \\ 1 \\ 1 \\ 0 \\ 1 \end{array}\right]\right\}\).


Example 38

V7 - Basis Computation (ver. 38)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} -2 \\ 1 \\ 5 \end{array}\right]\right\}\).


Example 39

V7 - Basis Computation (ver. 39)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -2 \\ 0 \\ 3 \\ -3 \\ -4 \end{array}\right],\left[\begin{array}{r} 1 \\ 1 \\ -4 \\ 7 \\ 7 \end{array}\right]\right\}\).


Example 40

V7 - Basis Computation (ver. 40)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 2 \\ 4 \\ 1 \\ 0 \end{array}\right],\left[\begin{array}{r} 3 \\ 5 \\ 2 \\ -3 \end{array}\right],\left[\begin{array}{r} 6 \\ 7 \\ 5 \\ -7 \end{array}\right]\right\}\).


Example 41

V7 - Basis Computation (ver. 41)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ -4 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 1 \end{array}\right]\right\}\).


Example 42

V7 - Basis Computation (ver. 42)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ -4 \\ -3 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ -1 \\ 1 \end{array}\right]\right\}\).


Example 43

V7 - Basis Computation (ver. 43)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -3 \\ 0 \\ 2 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ -2 \end{array}\right]\right\}\).


Example 44

V7 - Basis Computation (ver. 44)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} -1 \\ 1 \\ -2 \end{array}\right],\left[\begin{array}{r} 0 \\ 1 \\ 0 \end{array}\right]\right\}\).


Example 45

V7 - Basis Computation (ver. 45)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ 0 \\ 2 \\ -1 \end{array}\right],\left[\begin{array}{r} 3 \\ -2 \\ 0 \\ 4 \\ -1 \end{array}\right]\right\}\).


Example 46

V7 - Basis Computation (ver. 46)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ -1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ -3 \\ -3 \\ 2 \end{array}\right]\right\}\).


Example 47

V7 - Basis Computation (ver. 47)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 2 \\ -3 \\ 0 \\ 1 \end{array}\right],\left[\begin{array}{r} -3 \\ -5 \\ 5 \\ 4 \\ -4 \end{array}\right]\right\}\).


Example 48

V7 - Basis Computation (ver. 48)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 1 \\ 0 \\ 0 \end{array}\right],\left[\begin{array}{r} 2 \\ 1 \\ -1 \end{array}\right]\right\}\).


Example 49

V7 - Basis Computation (ver. 49)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 0 \\ 2 \\ -1 \\ -2 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ -1 \\ -1 \end{array}\right],\left[\begin{array}{r} 3 \\ 4 \\ 0 \\ 0 \end{array}\right]\right\}\).


Example 50

V7 - Basis Computation (ver. 50)

Find a basis for the subspace

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{r} 3 \\ 0 \\ 2 \\ 4 \end{array}\right],\left[\begin{array}{r} -1 \\ 1 \\ -1 \\ 0 \end{array}\right],\left[\begin{array}{r} 1 \\ 3 \\ 0 \\ 2 \end{array}\right]\right\}\).