## G4 - Eigenvectors

#### Example 1 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-4$$ in the matrix

$\left[\begin{array}{cccc} -6 & 8 & -6 & -4 \\ 1 & -8 & 3 & 2 \\ 1 & -4 & -1 & 2 \\ 1 & -4 & 3 & -2 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & 8 & -6 & -4 \\ 1 & -4 & 3 & 2 \\ 1 & -4 & 3 & 2 \\ 1 & -4 & 3 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & -4 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 2 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-2$$ in the matrix

$\left[\begin{array}{cccc} -1 & -1 & -4 & -4 \\ -1 & -4 & 1 & -2 \\ 0 & 4 & 2 & 8 \\ 0 & -2 & -2 & -6 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -4 & -4 \\ -1 & -2 & 1 & -2 \\ 0 & 4 & 4 & 8 \\ 0 & -2 & -2 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & -2 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 3 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 3 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$2$$ in the matrix

$\left[\begin{array}{cccc} 5 & -6 & 3 & -3 \\ 2 & -2 & 2 & -2 \\ -1 & 2 & 1 & 1 \\ -4 & 8 & -4 & 6 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 3 & -6 & 3 & -3 \\ 2 & -4 & 2 & -2 \\ -1 & 2 & -1 & 1 \\ -4 & 8 & -4 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 4 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 2 & 3 & 2 & -7 \\ -1 & -1 & -1 & 3 \\ -3 & -2 & -2 & 5 \\ 2 & 8 & 0 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -1 \\ 1 \\ 3 \\ 1 \end{array}\right] \right\}$$.

#### Example 5 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -3 & 3 & 0 & 6 \\ -1 & 1 & 0 & 2 \\ 2 & -2 & 0 & -4 \\ -4 & 4 & 0 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & -1 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 6 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$2$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ 0 & 1 & -1 & 5 \\ 1 & 0 & 2 & -7 \\ 3 & 2 & 1 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ -2 \\ 3 \\ 1 \end{array}\right] \right\}$$.

#### Example 7 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$3$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & -1 & -1 \\ 1 & 0 & 0 & 1 \\ 5 & -2 & -2 & 1 \\ 2 & -3 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 0 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 8 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -5 & 6 \\ 1 & -2 & -5 & 6 \\ -1 & 3 & 6 & -8 \\ 0 & -1 & -1 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & 2 \\ 0 & 1 & 1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 3 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 9 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -1 & 0 & -1 \\ 1 & 0 & 1 & 1 \\ -2 & 0 & -2 & -2 \\ -1 & 1 & -2 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -1 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 10 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 1 & 3 & 2 \\ -1 & 1 & -1 & 1 \\ 3 & -3 & 4 & -2 \\ 3 & -1 & 5 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 3 \\ 1 \\ -1 \\ 1 \end{array}\right] \right\}$$.

#### Example 11 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

$\left[\begin{array}{cccc} 2 & 1 & 0 & -4 \\ 5 & 0 & 5 & 8 \\ 3 & -2 & 6 & 5 \\ 0 & 1 & -7 & 9 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & 1 & 0 & -4 \\ 5 & -4 & 5 & 8 \\ 3 & -2 & 2 & 5 \\ 0 & 1 & -7 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ 1 \end{array}\right] \right\}$$.

#### Example 12 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & -2 & -4 & -2 \\ 1 & 3 & 4 & 5 \\ -1 & 2 & 6 & 0 \\ -1 & -4 & -6 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 2 \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 13 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$2$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 3 & -2 & 3 \\ 1 & -2 & 1 & -3 \\ -1 & 6 & -5 & 3 \\ -2 & 6 & -4 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -3 \\ 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 14 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$3$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -4 & 6 & -8 \\ 0 & 0 & 1 & -1 \\ -2 & 8 & 3 & 1 \\ 0 & 0 & -5 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & -4 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 1 \end{array}\right] \right\}$$.

#### Example 15 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & 0 & 0 & -2 \\ 3 & 0 & 0 & 3 \\ 5 & 0 & 0 & 5 \\ -5 & 0 & 0 & -5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 16 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -3 & 5 & 7 \\ 1 & 2 & -4 & -5 \\ 0 & -3 & 3 & 6 \\ -2 & -2 & 6 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -1 \\ 0 & 1 & -1 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 17 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 4 & 1 & 2 & 6 \\ 3 & 1 & 2 & 5 \\ 0 & 3 & 6 & 6 \\ 5 & -4 & -8 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 0 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 18 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & 2 & 2 \\ -1 & 2 & 6 & 6 \\ -3 & 1 & 8 & 8 \\ 0 & 1 & 2 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -2 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 19 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & 3 & 0 \\ -1 & 3 & -1 & 3 \\ -1 & 3 & 1 & 3 \\ 1 & -3 & 2 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & -3 & 0 & -3 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 3 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 20 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-1$$ in the matrix

$\left[\begin{array}{cccc} 6 & 7 & 0 & 7 \\ -4 & -5 & 0 & -4 \\ 3 & 3 & -1 & 3 \\ 4 & 4 & 0 & 3 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 7 & 7 & 0 & 7 \\ -4 & -4 & 0 & -4 \\ 3 & 3 & 0 & 3 \\ 4 & 4 & 0 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 21 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

$\left[\begin{array}{cccc} -2 & 4 & -5 & -1 \\ 1 & 1 & -4 & -1 \\ -1 & -4 & -2 & 1 \\ -2 & -8 & 5 & -1 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & -5 & -1 \\ 1 & 4 & -4 & -1 \\ -1 & -4 & 1 & 1 \\ -2 & -8 & 5 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 4 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 22 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$2$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -3 & 1 & 3 \\ -1 & -2 & 1 & 2 \\ -2 & -4 & 3 & 1 \\ -2 & -6 & 3 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 3 \\ 1 \\ 3 \\ 1 \end{array}\right] \right\}$$.

#### Example 23 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 2 & 1 & 2 & 2 \\ -1 & 0 & -1 & -2 \\ 0 & 2 & 0 & -4 \\ -4 & -5 & -4 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 2 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 24 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 4 & 1 & 3 & 6 \\ -2 & -1 & -1 & -4 \\ -3 & 1 & -4 & -1 \\ 1 & 0 & 1 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -1 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 25 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -1 & 3 & 0 \\ 1 & 0 & -1 & -2 \\ 0 & 0 & 0 & 0 \\ -1 & -1 & 3 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -2 \\ 0 & 1 & -2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ 2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 26 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ -2 & -8 & 1 & -3 \\ -1 & -4 & 0 & -3 \\ 2 & 8 & -1 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 4 & 0 & 3 \\ 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ -3 \\ 1 \end{array}\right] \right\}$$.

#### Example 27 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$2$$ in the matrix

$\left[\begin{array}{cccc} -2 & 7 & -3 & 2 \\ 3 & 0 & -1 & 5 \\ 5 & -8 & 5 & -1 \\ 0 & 4 & -4 & 10 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} -4 & 7 & -3 & 2 \\ 3 & -2 & -1 & 5 \\ 5 & -8 & 3 & -1 \\ 0 & 4 & -4 & 8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 3 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 28 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 2 & 6 & 4 & 4 \\ -1 & -3 & -2 & -2 \\ 0 & 0 & 0 & 0 \\ -1 & -3 & -2 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 3 & 2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -3 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 29 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 1 & -2 & -2 & -3 \\ 0 & 0 & 0 & 0 \\ 2 & -4 & -4 & -6 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & -2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 30 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -3 & 5 & 8 \\ 0 & 1 & -2 & -3 \\ 1 & -2 & 3 & 5 \\ 0 & 1 & -2 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & -1 \\ 0 & 1 & -2 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ 2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 31 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & -1 & 1 \\ -4 & -3 & 4 & -3 \\ 3 & -2 & -3 & -2 \\ 5 & 0 & -5 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 32 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-4$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 2 & 5 \\ 0 & 1 & 4 & 6 \\ 0 & 1 & 5 & 8 \\ 0 & -1 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 3 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -3 \\ 2 \\ -2 \\ 1 \end{array}\right] \right\}$$.

#### Example 33 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 2 & 0 & 1 \\ 0 & 1 & 1 & 1 \\ 1 & -3 & -1 & -2 \\ 0 & -1 & -1 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 2 & 1 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -2 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 34 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-2$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 5 & 4 \\ -1 & 1 & -6 & -5 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 0 \\ -1 \\ -1 \\ 1 \end{array}\right] \right\}$$.

#### Example 35 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-2$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -2 & -3 & -6 & -4 \\ -1 & -2 & -4 & -3 \\ 0 & 0 & 0 & 0 \\ -2 & -3 & -6 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -1 \\ 0 & 1 & 2 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 0 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 36 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-2$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 4 & 2 & 3 \\ 0 & 1 & 0 & 2 \\ 0 & -3 & 1 & -7 \\ -2 & -5 & -2 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -3 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 3 \\ -2 \\ 1 \\ 1 \end{array}\right] \right\}$$.

#### Example 37 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$3$$ in the matrix

$\left[\begin{array}{cccc} 2 & -2 & -3 & -2 \\ 2 & 7 & 6 & 4 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -2 & -3 & -2 \\ 2 & 4 & 6 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 2 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 38 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -3 & 7 & 7 \\ 0 & 1 & -3 & -4 \\ 0 & 2 & -6 & -7 \\ 0 & 1 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 0 \\ 0 & 1 & -3 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ 3 \\ 1 \\ 0 \end{array}\right] \right\}$$.

#### Example 39 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 5 & -1 & 2 \\ 1 & -5 & 0 & -1 \\ -1 & 5 & -2 & 3 \\ 1 & -5 & -6 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & -5 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 5 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 1 \\ 1 \end{array}\right] \right\}$$.

#### Example 40 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

$\left[\begin{array}{cccc} -2 & -1 & 2 & 4 \\ -2 & -1 & -4 & -8 \\ 1 & -1 & -1 & 4 \\ 1 & -1 & 2 & 1 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 2 & 4 \\ -2 & 2 & -4 & -8 \\ 1 & -1 & 2 & 4 \\ 1 & -1 & 2 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & -1 & 2 & 4 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 41 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -1 & 0 & -1 \\ 4 & -3 & 3 & -8 \\ 1 & 0 & 4 & -6 \\ -1 & 1 & 3 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & -1 \\ 0 & 0 & 1 & -1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ 1 \end{array}\right] \right\}$$.

#### Example 42 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$3$$ in the matrix

$\left[\begin{array}{cccc} 4 & 2 & -2 & -2 \\ 0 & 3 & 0 & 0 \\ -2 & -4 & 7 & 4 \\ 4 & 8 & -8 & -5 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 2 & -2 & -2 \\ 0 & 0 & 0 & 0 \\ -2 & -4 & 4 & 4 \\ 4 & 8 & -8 & -8 \end{array}\right] = \left[\begin{array}{cccc} 1 & 2 & -2 & -2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 43 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-3$$ in the matrix

$\left[\begin{array}{cccc} -4 & 3 & -2 & 2 \\ -2 & 2 & -8 & -6 \\ -1 & -1 & -6 & -8 \\ -2 & 5 & -6 & -5 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & 3 & -2 & 2 \\ -2 & 5 & -8 & -6 \\ -1 & -1 & -3 & -8 \\ -2 & 5 & -6 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 0 \\ -2 \\ -2 \\ 1 \end{array}\right] \right\}$$.

#### Example 44 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-2$$ in the matrix

$\left[\begin{array}{cccc} 5 & 7 & 7 & 7 \\ -3 & -5 & -3 & -3 \\ -3 & -3 & -5 & -3 \\ 5 & 5 & 5 & 3 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 7 & 7 & 7 & 7 \\ -3 & -3 & -3 & -3 \\ -3 & -3 & -3 & -3 \\ 5 & 5 & 5 & 5 \end{array}\right] = \left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 45 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 3 & 2 & -1 \\ 0 & 0 & 0 & 0 \\ -1 & -3 & -2 & 1 \\ 2 & 6 & 4 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 3 & 2 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -3 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 46 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$3$$ in the matrix

$\left[\begin{array}{cccc} 6 & 3 & 3 & 3 \\ -1 & 2 & -1 & -1 \\ 5 & 5 & 8 & 5 \\ 3 & 3 & 3 & 6 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 3 & 3 & 3 & 3 \\ -1 & -1 & -1 & -1 \\ 5 & 5 & 5 & 5 \\ 3 & 3 & 3 & 3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} -1 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 47 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$2$$ in the matrix

$\left[\begin{array}{cccc} 3 & -2 & -2 & 1 \\ -4 & 10 & 8 & -4 \\ 2 & -4 & -2 & 2 \\ 4 & -8 & -8 & 6 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -2 & -2 & 1 \\ -4 & 8 & 8 & -4 \\ 2 & -4 & -4 & 2 \\ 4 & -8 & -8 & 4 \end{array}\right] = \left[\begin{array}{cccc} 1 & -2 & -2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 48 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 1 & -2 \\ 0 & 1 & 3 & -1 \\ 0 & -1 & -3 & 1 \\ 1 & 1 & 1 & -2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -1 \\ 0 & 1 & 3 & -1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ -3 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 49 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$-1$$ in the matrix

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

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & -7 & -8 & 2 \\ 0 & 1 & 1 & 0 \\ 2 & -4 & -6 & 4 \\ 1 & -5 & -6 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -1 & 2 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 1 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 0 \\ 1 \end{array}\right] \right\}$$.

#### Example 50 π

Explain how to find a basis for the eigenspace associated to the eigenvalue $$4$$ in the matrix

$\left[\begin{array}{cccc} 5 & 1 & 2 & 1 \\ 0 & 5 & 4 & 5 \\ 0 & 0 & 5 & 1 \\ 0 & 1 & 5 & 10 \end{array}\right]$

$\operatorname{RREF} \left[\begin{array}{cccc} 1 & 1 & 2 & 1 \\ 0 & 1 & 4 & 5 \\ 0 & 0 & 1 & 1 \\ 0 & 1 & 5 & 6 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 0 & -2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$
A basis of the eigenspace is $$\left\{ \left[\begin{array}{c} 2 \\ -1 \\ -1 \\ 1 \end{array}\right] \right\}$$.