V9 - Polynomial Computation


Example 1

V9 - Polynomial Computation (ver. 1)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{4 \, x^{3} - 2 \, x^{2} - x + 2,-2 \, x^{3} - x^{2} + x,x^{3} + 2 \, x^{2} - x - 2,-5 \, x^{3} + 6 \, x^{2} - 8\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{4 \, x^{3} - 2 \, x^{2} - x + 2,-2 \, x^{3} - x^{2} + x,x^{3} + 2 \, x^{2} - x - 2\right\}\).


Example 2

V9 - Polynomial Computation (ver. 2)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-x^{3} - 2 \, x^{2} - x,x^{2} + 2,-3 \, x^{3} - 6 \, x^{2} - 4 \, x + 6,x^{3} + x^{2} + x - 2\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-x^{3} - 2 \, x^{2} - x,x^{2} + 2,-3 \, x^{3} - 6 \, x^{2} - 4 \, x + 6\right\}\).


Example 3

V9 - Polynomial Computation (ver. 3)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 4

V9 - Polynomial Computation (ver. 4)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-3 \, x^{3} + 2 \, x^{2} + 3,x^{3} - x^{2} + 2 \, x - 2,-6 \, x^{3} + 5 \, x^{2} - 5 \, x + 6,-x^{3} + 4 \, x - 1\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-3 \, x^{3} + 2 \, x^{2} + 3,x^{3} - x^{2} + 2 \, x - 2,-6 \, x^{3} + 5 \, x^{2} - 5 \, x + 6\right\}\).


Example 5

V9 - Polynomial Computation (ver. 5)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} + 4 \, x + 3,x^{2} - 4 \, x - 4,-x^{3} - 4 \, x - 3,-2 \, x^{3} - 2 \, x^{2} + 2\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} + 4 \, x + 3,x^{2} - 4 \, x - 4\right\}\).


Example 6

V9 - Polynomial Computation (ver. 6)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 7

V9 - Polynomial Computation (ver. 7)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-x^{3} - x^{2} - x - 3,5 \, x^{3} + 4 \, x^{2} + x + 6,2 \, x^{3} + x^{2} - x + 1,3 \, x^{3} + x^{2} - 3 \, x - 1\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-x^{3} - x^{2} - x - 3,5 \, x^{3} + 4 \, x^{2} + x + 6,2 \, x^{3} + x^{2} - x + 1\right\}\).


Example 8

V9 - Polynomial Computation (ver. 8)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-3 \, x^{3} + x^{2} + 2,-x^{2} - 2 \, x - 1,5 \, x^{3} - x - 3,5 \, x^{3} - 3 \, x^{2} - 7 \, x - 6\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-3 \, x^{3} + x^{2} + 2,-x^{2} - 2 \, x - 1,5 \, x^{3} - x - 3\right\}\).


Example 9

V9 - Polynomial Computation (ver. 9)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{3 \, x^{3} - 2 \, x^{2} + x - 5,x^{2},2 \, x^{3} + 5 \, x^{2} + x + 3,-5 \, x^{3} - 6 \, x^{2} - 2 \, x + 2\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{3 \, x^{3} - 2 \, x^{2} + x - 5,x^{2},2 \, x^{3} + 5 \, x^{2} + x + 3\right\}\).


Example 10

V9 - Polynomial Computation (ver. 10)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 11

V9 - Polynomial Computation (ver. 11)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 12

V9 - Polynomial Computation (ver. 12)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-x^{2} - 2 \, x,x^{3} - 2 \, x^{2} - 2 \, x + 2,-2 \, x^{3} + 3 \, x^{2} + 2 \, x - 4,x^{3} - 2 \, x^{2} - 2 \, x + 2\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-x^{2} - 2 \, x,x^{3} - 2 \, x^{2} - 2 \, x + 2\right\}\).


Example 13

V9 - Polynomial Computation (ver. 13)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{5 \, x^{3} + 2 \, x^{2} - 4 \, x - 4,7 \, x^{3} + 3 \, x^{2} - 4 \, x - 8,8 \, x^{3} + 3 \, x^{2} - 8 \, x - 4,-5 \, x^{3} - 2 \, x^{2} + 4 \, x + 4\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{5 \, x^{3} + 2 \, x^{2} - 4 \, x - 4,7 \, x^{3} + 3 \, x^{2} - 4 \, x - 8\right\}\).


Example 14

V9 - Polynomial Computation (ver. 14)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} - 2 \, x^{2} - x,-3 \, x^{2} - x - 2,6 \, x^{2} + 2 \, x + 4,-x^{3} - 7 \, x^{2} - 2 \, x - 6\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} - 2 \, x^{2} - x,-3 \, x^{2} - x - 2\right\}\).


Example 15

V9 - Polynomial Computation (ver. 15)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 16

V9 - Polynomial Computation (ver. 16)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 17

V9 - Polynomial Computation (ver. 17)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 18

V9 - Polynomial Computation (ver. 18)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{2} + 2 \, x,-x^{3} - 2 \, x + 2,-4 \, x^{3} + 2 \, x^{2} - 3 \, x + 7,5 \, x^{3} - 7 \, x^{2} - 6 \, x - 8\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{2} + 2 \, x,-x^{3} - 2 \, x + 2,-4 \, x^{3} + 2 \, x^{2} - 3 \, x + 7\right\}\).


Example 19

V9 - Polynomial Computation (ver. 19)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 20

V9 - Polynomial Computation (ver. 20)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} + x^{2} - 2 \, x,x^{3} + 2 \, x^{2} - 2,3 \, x^{3} + 3 \, x^{2} - 6 \, x,2 \, x^{3} + 5 \, x^{2} + 2 \, x - 6\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} + x^{2} - 2 \, x,x^{3} + 2 \, x^{2} - 2\right\}\).


Example 21

V9 - Polynomial Computation (ver. 21)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} - x + 1,-3 \, x^{3} + x^{2} + 2 \, x - 3,-2 \, x^{3} + x^{2} + x - 2,7 \, x^{3} - 2 \, x^{2} - 5 \, x + 7\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} - x + 1,-3 \, x^{3} + x^{2} + 2 \, x - 3\right\}\).


Example 22

V9 - Polynomial Computation (ver. 22)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-2 \, x^{3} - 2 \, x^{2} - 3 \, x + 2,2 \, x^{3} - x^{2} + x - 1,4 \, x^{3} + x^{2} + 4 \, x - 3,-6 \, x^{2} - 4 \, x + 2\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-2 \, x^{3} - 2 \, x^{2} - 3 \, x + 2,2 \, x^{3} - x^{2} + x - 1\right\}\).


Example 23

V9 - Polynomial Computation (ver. 23)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-x^{3} - x^{2} - x,3 \, x^{3} + 3 \, x^{2} + 3 \, x,7 \, x^{3} + 6 \, x^{2} + 6 \, x + 4,8 \, x^{3} + 7 \, x^{2} + 7 \, x + 4\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-x^{3} - x^{2} - x,7 \, x^{3} + 6 \, x^{2} + 6 \, x + 4\right\}\).


Example 24

V9 - Polynomial Computation (ver. 24)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-x^{3} - 2 \, x^{2} - 2,-4 \, x^{3} - 8 \, x^{2} - 8,3 \, x^{3} + x^{2} + 3,-6 \, x^{3} - 2 \, x^{2} - 6\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-x^{3} - 2 \, x^{2} - 2,3 \, x^{3} + x^{2} + 3\right\}\).


Example 25

V9 - Polynomial Computation (ver. 25)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-x^{3} + 2 \, x^{2} - 2 \, x - 1,-4 \, x^{3} + 8 \, x^{2} - 8 \, x - 4,-2 \, x^{3} + x^{2} - 4 \, x - 1,x^{3} + 4 \, x^{2} + 2 \, x - 1\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-x^{3} + 2 \, x^{2} - 2 \, x - 1,-2 \, x^{3} + x^{2} - 4 \, x - 1\right\}\).


Example 26

V9 - Polynomial Computation (ver. 26)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-2 \, x^{3} - x^{2} - x + 1,x^{2} + x,-5 \, x^{3} + x + 3,-7 \, x^{3} + x^{2} + 2 \, x + 4\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-2 \, x^{3} - x^{2} - x + 1,x^{2} + x,-5 \, x^{3} + x + 3\right\}\).


Example 27

V9 - Polynomial Computation (ver. 27)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-2 \, x^{3} - x^{2} + 2 \, x + 1,-x^{3} - x^{2} + 3 \, x - 1,4 \, x^{3} + 3 \, x^{2} - 8 \, x + 1,5 \, x^{3} + 3 \, x^{2} - 7 \, x - 1\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-2 \, x^{3} - x^{2} + 2 \, x + 1,-x^{3} - x^{2} + 3 \, x - 1\right\}\).


Example 28

V9 - Polynomial Computation (ver. 28)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-7 \, x^{3} - 2 \, x^{2} - 2 \, x + 4,5 \, x^{3} + 3 \, x^{2} + x - 3,-8 \, x^{3} - 7 \, x^{2} - x + 5,-2 \, x^{3} + x^{2} - x + 1\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-7 \, x^{3} - 2 \, x^{2} - 2 \, x + 4,5 \, x^{3} + 3 \, x^{2} + x - 3\right\}\).


Example 29

V9 - Polynomial Computation (ver. 29)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} + 2 \, x^{2} - 3 \, x + 1,3 \, x^{3} + 7 \, x^{2} - 8 \, x + 7,x^{2} + 2 \, x,3 \, x^{3} + 8 \, x^{2} - 6 \, x + 7\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} + 2 \, x^{2} - 3 \, x + 1,3 \, x^{3} + 7 \, x^{2} - 8 \, x + 7,x^{2} + 2 \, x\right\}\).


Example 30

V9 - Polynomial Computation (ver. 30)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{4 \, x^{3} + 5 \, x - 3,3 \, x^{3} + x^{2} - 4 \, x - 3,4 \, x^{3} + 5 \, x - 3,4 \, x^{3} + 5 \, x - 3\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{4 \, x^{3} + 5 \, x - 3,3 \, x^{3} + x^{2} - 4 \, x - 3\right\}\).


Example 31

V9 - Polynomial Computation (ver. 31)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} + 5 \, x^{2} + 5 \, x + 1,-x^{3} - 5 \, x^{2} - 5 \, x - 1,-x^{3} - 4 \, x^{2} - 5 \, x - 6,-x^{3} - 6 \, x^{2} - 5 \, x + 4\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} + 5 \, x^{2} + 5 \, x + 1,-x^{3} - 4 \, x^{2} - 5 \, x - 6\right\}\).


Example 32

V9 - Polynomial Computation (ver. 32)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{rr} -1 & -1 \\ -1 & 1 \end{array}\right],\left[\begin{array}{rr} 3 & 0 \\ -2 & -1 \end{array}\right],\left[\begin{array}{rr} 8 & -1 \\ -6 & -2 \end{array}\right]\right\}\).


Example 33

V9 - Polynomial Computation (ver. 33)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} - x,-x^{3} + x^{2} + 2 \, x + 1,3 \, x^{3} - 2 \, x^{2} - 5 \, x - 2,-x^{3} + 3 \, x^{2} + 4 \, x + 3\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} - x,-x^{3} + x^{2} + 2 \, x + 1\right\}\).


Example 34

V9 - Polynomial Computation (ver. 34)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} + x,x^{2} - x - 1,-x^{3} + 3 \, x^{2} - 3 \, x - 2,-x^{3} + 4 \, x^{2} - 4 \, x - 3\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} + x,x^{2} - x - 1,-x^{3} + 3 \, x^{2} - 3 \, x - 2\right\}\).


Example 35

V9 - Polynomial Computation (ver. 35)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 36

V9 - Polynomial Computation (ver. 36)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-x^{3} - x^{2} - 1,x^{3} + x,x^{3} + x^{2} - 3 \, x + 3,-2 \, x^{3} + x^{2} + 3 \, x - 3\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-x^{3} - x^{2} - 1,x^{3} + x,x^{3} + x^{2} - 3 \, x + 3\right\}\).


Example 37

V9 - Polynomial Computation (ver. 37)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-2 \, x^{3} - x^{2} + 2 \, x + 3,2 \, x^{2} - 1,-3 \, x^{3} - x^{2} + 3 \, x + 4,-4 \, x^{3} - 7 \, x^{2} + 4 \, x + 8\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-2 \, x^{3} - x^{2} + 2 \, x + 3,2 \, x^{2} - 1,-3 \, x^{3} - x^{2} + 3 \, x + 4\right\}\).


Example 38

V9 - Polynomial Computation (ver. 38)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} + x^{2} + 1,-3 \, x^{3} - 2 \, x^{2} - 6,4 \, x^{3} + 3 \, x^{2} + 7,-3 \, x^{3} - 3 \, x^{2} - 3\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} + x^{2} + 1,-3 \, x^{3} - 2 \, x^{2} - 6\right\}\).


Example 39

V9 - Polynomial Computation (ver. 39)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-2 \, x^{2} - x - 1,-3 \, x^{3} - 7 \, x^{2} - 7 \, x - 8,x^{3} + 4 \, x^{2} + 3 \, x + 5,x^{3} - x^{2} + x - 2\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-2 \, x^{2} - x - 1,-3 \, x^{3} - 7 \, x^{2} - 7 \, x - 8,x^{3} + 4 \, x^{2} + 3 \, x + 5\right\}\).


Example 40

V9 - Polynomial Computation (ver. 40)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} - 1,3 \, x^{3} + x^{2} + 2 \, x - 2,8 \, x^{3} + 3 \, x^{2} + 6 \, x - 5,-4 \, x^{3} - x^{2} - 2 \, x + 3\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} - 1,3 \, x^{3} + x^{2} + 2 \, x - 2\right\}\).


Example 41

V9 - Polynomial Computation (ver. 41)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 42

V9 - Polynomial Computation (ver. 42)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 43

V9 - Polynomial Computation (ver. 43)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-2 \, x^{3} + 3 \, x^{2} + 2,x^{3} - 2 \, x^{2} - 2 \, x,-x^{3} - 6 \, x + 4,-5 \, x^{3} + 7 \, x^{2} - 2 \, x + 6\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-2 \, x^{3} + 3 \, x^{2} + 2,x^{3} - 2 \, x^{2} - 2 \, x\right\}\).


Example 44

V9 - Polynomial Computation (ver. 44)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 45

V9 - Polynomial Computation (ver. 45)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{-2 \, x^{3} - 3 \, x^{2} + 2 \, x - 3,-2 \, x^{3} - 5 \, x^{2} + 2 \, x - 4,-6 \, x^{2} + x - 3,6 \, x^{2} + 3\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{-2 \, x^{3} - 3 \, x^{2} + 2 \, x - 3,-2 \, x^{3} - 5 \, x^{2} + 2 \, x - 4,-6 \, x^{2} + x - 3\right\}\).


Example 46

V9 - Polynomial Computation (ver. 46)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 47

V9 - Polynomial Computation (ver. 47)

Find a basis for the subspace of \(\mathcal P_3\) given by

\[W=\mathrm{span}\left\{x^{3} + x - 1,3 \, x^{3} + x^{2} + 3 \, x - 5,-7 \, x^{3} - x^{2} - 6 \, x + 7,7 \, x^{3} + 5 \, x - 3\right\}.\]

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{x^{3} + x - 1,3 \, x^{3} + x^{2} + 3 \, x - 5,-7 \, x^{3} - x^{2} - 6 \, x + 7\right\}\).


Example 48

V9 - Polynomial Computation (ver. 48)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

A basis of \(W\) is \(\left\{\left[\begin{array}{rr} 1 & -1 \\ 0 & 4 \end{array}\right],\left[\begin{array}{rr} -5 & 4 \\ -1 & -8 \end{array}\right]\right\}\).


Example 49

V9 - Polynomial Computation (ver. 49)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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


Example 50

V9 - Polynomial Computation (ver. 50)

Find a basis for the subspace of \(M_{2,2}\) given by

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

Be sure to explain why your result is a basis.

Answer.

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

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