## V7 - Basis of a subspace

#### Example 1 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 3 \\ 3 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 9 \\ 9 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -21 \\ -21 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -44 \\ -44 \\ -7 \\ 8 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} 3 & -2 & 9 & -21 & -44 \\ 3 & -2 & 9 & -21 & -44 \\ -1 & -1 & 2 & -3 & -7 \\ 3 & 2 & -3 & 3 & 8 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 1 & -3 & -6 \\ 0 & 1 & -3 & 6 & 13 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. A basis of $$W$$ is $$\left\{ \left[\begin{array}{c} 3 \\ 3 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -1 \\ 2 \end{array}\right] \right\}$$.
2. The dimension of $$W$$ is $$2$$.

#### Example 2 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

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

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

#### Example 3 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -1 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 5 \\ -15 \\ -15 \\ 15 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -3 \\ 2 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -2 & 5 & -2 \\ -1 & 3 & -15 & 1 \\ -1 & 3 & -15 & -3 \\ 1 & -3 & 15 & 2 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 3 & 0 \\ 0 & 1 & -4 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

1. A basis of $$W$$ is $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -3 \\ 2 \end{array}\right] \right\}$$.
2. The dimension of $$W$$ is $$3$$.

#### Example 4 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} -3 & -4 & 1 & -2 & 2 \\ -4 & -4 & 3 & 0 & -3 \\ 2 & -2 & -3 & -1 & 3 \\ 1 & -2 & 3 & 3 & 0 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & \frac{19}{7} \\ 0 & 1 & 0 & 0 & -\frac{5}{14} \\ 0 & 0 & 1 & 0 & \frac{15}{7} \\ 0 & 0 & 0 & 1 & -\frac{23}{7} \end{array}\right]$

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

#### Example 5 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 0 \\ 0 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -8 \\ 12 \\ -18 \\ -16 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 30 \\ -45 \\ 56 \\ 52 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ -1 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} 0 & -2 & -8 & -2 & 30 & 2 \\ 0 & 3 & 12 & 3 & -45 & -4 \\ 2 & -4 & -18 & 2 & 56 & 1 \\ 0 & -4 & -16 & 0 & 52 & -1 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -1 & 0 & 4 & 0 \\ 0 & 1 & 4 & 0 & -13 & 0 \\ 0 & 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]$

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

#### Example 6 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

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

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

#### Example 7 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 3 \\ 1 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -11 \\ -1 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -3 \\ -3 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} 3 & -1 & -11 & -1 \\ 1 & 1 & -1 & -3 \\ 3 & 2 & -5 & -3 \\ 2 & 3 & 0 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -3 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

1. A basis of $$W$$ is $$\left\{ \left[\begin{array}{c} 3 \\ 1 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -3 \\ -3 \end{array}\right] \right\}$$.
2. The dimension of $$W$$ is $$3$$.

#### Example 8 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -4 \\ 0 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 6 \\ -14 \\ -2 \\ -6 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} -4 & 3 & -1 & -1 & 6 \\ 0 & 3 & 0 & -4 & -14 \\ 2 & -1 & -2 & -1 & -2 \\ 2 & -3 & 1 & -2 & -6 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & -3 \\ 0 & 1 & 0 & 0 & -2 \\ 0 & 0 & 1 & 0 & -2 \\ 0 & 0 & 0 & 1 & 2 \end{array}\right]$

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

#### Example 9 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 15 \\ 10 \\ -12 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ 1 \\ 0 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -3 & 3 & 15 & -3 & 1 & -2 \\ -3 & -2 & 10 & 2 & -4 & 2 \\ 2 & -4 & -12 & -1 & -2 & 1 \\ 0 & -2 & -2 & 0 & 3 & 0 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -4 & 0 & 0 & -\frac{6}{355} \\ 0 & 1 & 1 & 0 & 0 & -\frac{93}{355} \\ 0 & 0 & 0 & 1 & 0 & \frac{129}{355} \\ 0 & 0 & 0 & 0 & 1 & -\frac{62}{355} \end{array}\right]$

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

#### Example 10 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 2 \\ 2 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -14 \\ -13 \\ -18 \\ 12 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} 2 & 3 & 1 & -1 & 0 & -14 \\ 2 & 1 & 3 & -4 & 2 & -13 \\ 3 & 2 & 1 & 2 & -3 & -18 \\ 0 & 2 & -4 & -4 & -3 & 12 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -\frac{241}{142} & -3 \\ 0 & 1 & 0 & 0 & \frac{89}{142} & -2 \\ 0 & 0 & 1 & 0 & \frac{183}{142} & -3 \\ 0 & 0 & 0 & 1 & -\frac{16}{71} & -1 \end{array}\right]$

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

#### Example 11 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

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

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

#### Example 12 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 6 \\ -13 \\ 3 \end{array}\right] , \left[\begin{array}{c} -8 \\ -18 \\ 35 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ -2 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -4 & -2 & 1 & 1 & -8 & -4 \\ -2 & -2 & -2 & 6 & -18 & 1 \\ -2 & 3 & 2 & -13 & 35 & 2 \\ 3 & 0 & 0 & 3 & -6 & -2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 1 & -2 & 0 \\ 0 & 1 & 0 & -3 & 9 & 0 \\ 0 & 0 & 1 & -1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]$

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

#### Example 13 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 2 \\ -2 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 6 \\ 2 \\ -8 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -4 \\ 16 \end{array}\right] , \left[\begin{array}{c} -10 \\ 10 \\ -1 \\ 4 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} 2 & 0 & -6 & 4 & -10 \\ -2 & 0 & 6 & -4 & 10 \\ 0 & 1 & 2 & -4 & -1 \\ 0 & -4 & -8 & 16 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -3 & 2 & -5 \\ 0 & 1 & 2 & -4 & -1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 14 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 2 \\ 0 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ 28 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 15 \\ -101 \\ 18 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} 2 & 2 & 0 & -2 \\ 0 & -1 & -4 & 15 \\ -4 & 3 & 28 & -101 \\ -3 & -4 & -4 & 18 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -4 & 14 \\ 0 & 1 & 4 & -15 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 15 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -2 \\ 3 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -14 \\ -4 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 7 \\ -5 \\ -4 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} -2 & -4 & 3 & -14 & 1 \\ 3 & -1 & 1 & -4 & 7 \\ 0 & 2 & 1 & 2 & -5 \\ 2 & 3 & 2 & 2 & -4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 & -2 \\ 0 & 0 & 1 & -2 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 16 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 16 \\ 4 \\ 8 \\ 12 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -3 \\ 0 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} -4 & -4 & 16 & -1 \\ -3 & 1 & 4 & 0 \\ 0 & -4 & 8 & -3 \\ -2 & -4 & 12 & 0 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 17 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -4 \\ -1 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 8 \\ 6 \\ -14 \\ 14 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 0 \\ -3 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} -4 & 0 & 8 & -2 \\ -1 & -1 & 6 & -2 \\ 3 & 2 & -14 & 0 \\ 1 & -4 & 14 & -3 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & 0 \\ 0 & 1 & -4 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 18 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 3 \\ 3 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -17 \\ -17 \\ 26 \\ 18 \end{array}\right] , \left[\begin{array}{c} -37 \\ -37 \\ 58 \\ 42 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} 3 & 1 & -17 & -37 \\ 3 & 1 & -17 & -37 \\ -4 & -2 & 26 & 58 \\ -2 & -2 & 18 & 42 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -4 & -8 \\ 0 & 1 & -5 & -13 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

1. A basis of $$W$$ is $$\left\{ \left[\begin{array}{c} 3 \\ 3 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ -2 \\ -2 \end{array}\right] \right\}$$.
2. The dimension of $$W$$ is $$2$$.

#### Example 19 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -2 \\ -2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -6 \\ -16 \\ -10 \\ -13 \end{array}\right] , \left[\begin{array}{c} 11 \\ 9 \\ 11 \\ 2 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -2 & 3 & -3 & 1 & -6 & 11 \\ -2 & 3 & -1 & -3 & -16 & 9 \\ -4 & 3 & -1 & 1 & -10 & 11 \\ -2 & 1 & 2 & -3 & -13 & 2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & -2 & -3 & -1 \\ 0 & 1 & 0 & -3 & -9 & 2 \\ 0 & 0 & 1 & -2 & -5 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. A basis of $$W$$ is $$\left\{ \left[\begin{array}{c} -2 \\ -2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 2 \end{array}\right] \right\}$$.
2. The dimension of $$W$$ is $$3$$.

#### Example 20 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 3 \\ 0 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -3 \\ -10 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} 3 & -2 & 0 & -5 & -3 \\ 0 & 1 & -2 & -1 & -1 \\ -1 & 0 & -1 & 0 & -3 \\ 0 & -4 & 1 & -3 & -10 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & -1 & 1 \\ 0 & 1 & 0 & 1 & 3 \\ 0 & 0 & 1 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 21 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

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

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

#### Example 22 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -3 \\ 2 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ -1 \\ -4 \end{array}\right] , \left[\begin{array}{c} 12 \\ -13 \\ 1 \\ 18 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -2 \\ 1 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -3 & 2 & -4 & -3 & 12 & -3 \\ 2 & 2 & 3 & 0 & -13 & -1 \\ 0 & 1 & -2 & -1 & 1 & -2 \\ -4 & -2 & 2 & -4 & 18 & 1 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -3 & -\frac{53}{90} \\ 0 & 1 & 0 & 0 & -2 & -\frac{43}{90} \\ 0 & 0 & 1 & 0 & -1 & \frac{17}{45} \\ 0 & 0 & 0 & 1 & -1 & \frac{23}{30} \end{array}\right]$

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

#### Example 23 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -3 \\ 3 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -1 \\ -4 \end{array}\right] , \left[\begin{array}{c} 15 \\ -15 \\ 10 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -34 \\ 28 \\ -20 \\ -14 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} -3 & -1 & 15 & -4 & -34 \\ 3 & 0 & -15 & 2 & 28 \\ -2 & -1 & 10 & -1 & -20 \\ 0 & -4 & 0 & -3 & -14 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -5 & 0 & 8 \\ 0 & 1 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 & 2 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 24 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 0 \\ -2 \\ -2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 10 \\ 7 \\ 17 \\ 1 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} 0 & -1 & 1 & -4 & 10 \\ -2 & 3 & -2 & 1 & 7 \\ -2 & -4 & -4 & -1 & 17 \\ 2 & 3 & -2 & -1 & 1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & -3 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & -2 \\ 0 & 0 & 0 & 1 & -3 \end{array}\right]$

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

#### Example 25 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -1 & -3 & -4 & -1 & -3 & -1 \\ -2 & -1 & 2 & -4 & -3 & -4 \\ 3 & 3 & 0 & -4 & 2 & 0 \\ 0 & -3 & -6 & 1 & -2 & 3 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -2 & 0 & 0 & 7 \\ 0 & 1 & 2 & 0 & 0 & 11 \\ 0 & 0 & 0 & 1 & 0 & 6 \\ 0 & 0 & 0 & 0 & 1 & -15 \end{array}\right]$

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

#### Example 26 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} -7 \\ -1 \\ -4 \\ 14 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -4 \\ -1 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -3 & -7 & 0 \\ -1 & 3 & -1 & -4 \\ 0 & -4 & -4 & -4 \\ 3 & 2 & 14 & -1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 4 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 27 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -4 & 1 & -4 & -4 & -1 & 3 \\ 1 & 1 & 2 & -2 & -4 & -1 \\ 3 & -3 & -3 & -2 & 3 & -2 \\ -3 & -4 & -4 & 0 & -3 & -4 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & \frac{249}{161} & \frac{60}{161} \\ 0 & 1 & 0 & 0 & \frac{13}{7} & \frac{13}{7} \\ 0 & 0 & 1 & 0 & -\frac{365}{161} & -\frac{183}{161} \\ 0 & 0 & 0 & 1 & \frac{33}{23} & \frac{11}{23} \end{array}\right]$

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

#### Example 28 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

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

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

#### Example 29 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

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

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

#### Example 30 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 0 \\ -4 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ 6 \\ -9 \\ 13 \end{array}\right] , \left[\begin{array}{c} 8 \\ -3 \\ 1 \\ 20 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} 0 & -1 & -2 & 1 & 1 & 8 \\ -4 & 2 & 1 & -2 & 6 & -3 \\ 3 & 1 & -2 & 0 & -9 & 1 \\ -2 & -3 & -4 & 0 & 13 & 20 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -2 & -1 \\ 0 & 1 & 0 & 0 & -3 & -2 \\ 0 & 0 & 1 & 0 & 0 & -3 \\ 0 & 0 & 0 & 1 & -2 & 0 \end{array}\right]$

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

#### Example 31 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} -3 & 1 & -3 & -4 & 0 \\ -3 & -4 & -4 & 2 & 1 \\ -1 & 1 & 0 & 2 & -3 \\ -1 & -2 & 1 & 0 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & \frac{18}{43} \\ 0 & 1 & 0 & 0 & -\frac{48}{43} \\ 0 & 0 & 1 & 0 & \frac{8}{43} \\ 0 & 0 & 0 & 1 & -\frac{63}{86} \end{array}\right]$

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

#### Example 32 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 12 \\ -8 \\ 12 \\ 12 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} 1 & 2 & 2 & -2 & 3 & 12 \\ 1 & -2 & 3 & 2 & 3 & -8 \\ 3 & -3 & 0 & -4 & -4 & 12 \\ 1 & 2 & -3 & -2 & -1 & 12 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & 1 & 2 \\ 0 & 1 & 0 & 0 & \frac{39}{35} & 2 \\ 0 & 0 & 1 & 0 & \frac{4}{5} & 0 \\ 0 & 0 & 0 & 1 & \frac{32}{35} & -3 \end{array}\right]$

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

#### Example 33 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 2 \\ -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -11 \\ 8 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 3 \\ -1 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} 2 & -4 & 1 & -11 & -3 & 3 \\ -3 & 1 & -2 & 8 & -3 & 3 \\ -4 & -4 & -1 & -5 & -4 & 3 \\ 0 & -3 & 0 & -6 & -3 & -1 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & 0 & -\frac{86}{9} \\ 0 & 1 & 0 & 2 & 0 & \frac{97}{9} \\ 0 & 0 & 1 & -3 & 0 & \frac{305}{9} \\ 0 & 0 & 0 & 0 & 1 & -\frac{94}{9} \end{array}\right]$

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

#### Example 34 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

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

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

#### Example 35 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -4 \\ 2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ 1 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 16 \\ -4 \\ 2 \\ -9 \end{array}\right] , \left[\begin{array}{c} 11 \\ -7 \\ -4 \\ 1 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} -4 & 2 & -3 & 16 & 11 \\ 2 & 1 & -1 & -4 & -7 \\ 0 & 1 & 2 & 2 & -4 \\ 1 & -3 & 2 & -9 & 1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & -3 & -3 \\ 0 & 1 & 0 & 2 & -2 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 36 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -2 \\ 1 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 16 \\ -7 \\ 0 \\ -9 \end{array}\right] , \left[\begin{array}{c} -14 \\ 6 \\ 0 \\ 10 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ -1 \\ -4 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -2 & -4 & 3 & 16 & -14 & 0 \\ 1 & 3 & 0 & -7 & 6 & 1 \\ 0 & 3 & 3 & 0 & 0 & -1 \\ -1 & 3 & -2 & -9 & 10 & -4 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 1 & 0 & -2 & 2 & 0 \\ 0 & 0 & 1 & 2 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]$

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

#### Example 37 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 2 \\ -1 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -12 \\ -19 \end{array}\right] , \left[\begin{array}{c} 5 \\ -3 \\ -9 \\ -13 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 9 \\ 17 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} 2 & -1 & 5 & 5 & -1 \\ -1 & 0 & -4 & -3 & 3 \\ -3 & 0 & -12 & -9 & 9 \\ -4 & -1 & -19 & -13 & 17 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 4 & 3 & -3 \\ 0 & 1 & 3 & 1 & -5 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 38 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 2 \\ -3 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 5 \\ 9 \end{array}\right] , \left[\begin{array}{c} -3 \\ 7 \\ 9 \\ 15 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} 2 & 1 & -1 & -3 \\ -3 & -1 & 3 & 7 \\ -1 & 1 & 5 & 9 \\ 0 & 3 & 9 & 15 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & -2 & -4 \\ 0 & 1 & 3 & 5 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 39 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 0 \\ 0 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 16 \\ 16 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ -2 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} 0 & 2 & -3 & -3 & -1 \\ 0 & 1 & -2 & -1 & 0 \\ 3 & -3 & -1 & 16 & -2 \\ 3 & -3 & -1 & 16 & -2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 2 & -3 \\ 0 & 1 & 0 & -3 & -2 \\ 0 & 0 & 1 & -1 & -1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 40 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 3 \\ -1 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -8 \\ 6 \end{array}\right] , \left[\begin{array}{c} 4 \\ -6 \\ -23 \\ 16 \end{array}\right] , \left[\begin{array}{c} 6 \\ -6 \\ -18 \\ 12 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 1 \\ -3 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} 3 & -1 & -1 & 4 & 6 & -4 \\ -1 & 1 & -1 & -6 & -6 & 2 \\ 2 & 3 & -8 & -23 & -18 & 1 \\ -2 & -2 & 6 & 16 & 12 & -3 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -1 & -1 & 0 & 0 \\ 0 & 1 & -2 & -7 & -6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. A basis of $$W$$ is $$\left\{ \left[\begin{array}{c} 3 \\ -1 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 1 \\ -3 \end{array}\right] \right\}$$.
2. The dimension of $$W$$ is $$3$$.

#### Example 41 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -10 \\ -4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} -22 \\ -8 \\ -3 \\ 9 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccc} -1 & -3 & -10 & -22 \\ -1 & -1 & -4 & -8 \\ -3 & 0 & -3 & -3 \\ 2 & 1 & 5 & 9 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 1 \\ 0 & 1 & 3 & 7 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 42 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 26 \\ 8 \\ 22 \\ -21 \end{array}\right] , \left[\begin{array}{c} 2 \\ 8 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 10 \\ -8 \\ 14 \\ -6 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ 2 \\ -2 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -4 & -3 & 26 & 2 & 10 & -1 \\ 0 & -4 & 8 & 8 & -8 & 2 \\ -4 & -1 & 22 & -2 & 14 & 2 \\ 3 & 3 & -21 & -3 & -6 & -2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -5 & 1 & -4 & 0 \\ 0 & 1 & -2 & -2 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 43 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -1 & 1 & -2 & 2 & 7 & 2 \\ -2 & -3 & 3 & 2 & 3 & -3 \\ 2 & 1 & -1 & 0 & -5 & -2 \\ -2 & -3 & 2 & 2 & 6 & -2 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 0 & -3 & -\frac{12}{7} \\ 0 & 1 & 0 & 0 & -2 & \frac{3}{7} \\ 0 & 0 & 1 & 0 & -3 & -1 \\ 0 & 0 & 0 & 1 & 0 & -\frac{15}{14} \end{array}\right]$

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

#### Example 44 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 0 \\ 0 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -12 \\ 3 \\ 9 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -6 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ -4 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} 0 & 1 & 3 & -1 & -2 \\ 0 & -4 & -12 & 4 & 1 \\ 3 & 0 & 3 & -6 & 1 \\ 0 & 3 & 9 & -3 & -4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 1 & -2 & 0 \\ 0 & 1 & 3 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 45 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 1 \\ -2 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -3 \\ -9 \end{array}\right] , \left[\begin{array}{c} 17 \\ -16 \\ 7 \\ 33 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 2 \\ 1 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 1 & -5 & 17 & -1 \\ -2 & 1 & 4 & -16 & -3 \\ -1 & 3 & -3 & 7 & 2 \\ 3 & 0 & -9 & 33 & 1 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -3 & 11 & 0 \\ 0 & 1 & -2 & 6 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 46 🔗

Consider the subspace

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

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} 2 & 0 & 3 & 1 & -2 & 3 \\ 3 & 1 & 1 & -4 & 3 & -3 \\ -1 & -4 & 0 & 2 & -4 & 0 \\ -2 & 0 & -3 & -1 & -2 & 1 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & -\frac{46}{25} & 0 & -\frac{16}{25} \\ 0 & 1 & 0 & -\frac{1}{25} & 0 & \frac{29}{25} \\ 0 & 0 & 1 & \frac{39}{25} & 0 & \frac{19}{25} \\ 0 & 0 & 0 & 0 & 1 & -1 \end{array}\right]$

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

#### Example 47 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 12 \\ 23 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -4 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -32 \\ -66 \\ -3 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} 0 & 0 & 0 & -4 & 3 & 4 \\ -3 & -1 & 12 & -4 & -4 & -32 \\ -4 & 1 & 23 & -3 & 0 & -66 \\ 1 & 2 & 1 & 0 & 2 & -3 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & -5 & 0 & 0 & 15 \\ 0 & 1 & 3 & 0 & 0 & -9 \\ 0 & 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right]$

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

#### Example 48 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -1 \\ -4 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -3 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 8 \\ 4 \\ -9 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ 7 \\ -8 \\ 4 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{ccccc} -1 & -1 & 2 & 8 & 5 \\ -4 & -2 & -3 & 4 & 7 \\ 3 & 2 & 0 & -9 & -8 \\ -3 & 0 & -2 & -1 & 4 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & -1 & -2 \\ 0 & 1 & 0 & -3 & -1 \\ 0 & 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

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

#### Example 49 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -1 \\ -2 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -10 \\ -12 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 28 \\ 33 \\ 3 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -1 & 3 & -1 & -2 & 0 & 2 \\ -2 & -2 & -10 & -1 & -3 & 28 \\ -3 & 0 & -12 & 1 & 3 & 33 \\ 0 & -1 & -1 & -2 & 1 & 3 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 4 & 0 & 0 & -11 \\ 0 & 1 & 1 & 0 & 0 & -3 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \end{array}\right]$

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

#### Example 50 🔗

Consider the subspace

$W=\operatorname{span} \left\{ \left[\begin{array}{c} -3 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -8 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 8 \\ 0 \\ -10 \end{array}\right] , \left[\begin{array}{c} 13 \\ -8 \\ -3 \\ 1 \end{array}\right] \right\} .$

1. Explain how to find a basis of $$W$$.
2. Explain how to find the dimension of $$W$$.

$\operatorname{RREF} \left[\begin{array}{cccccc} -3 & 2 & 0 & 1 & -4 & 13 \\ 0 & -4 & 0 & -8 & 8 & -8 \\ 0 & -2 & -1 & -2 & 0 & -3 \\ 0 & -1 & -3 & 4 & -10 & 1 \end{array}\right] = \left[\begin{array}{cccccc} 1 & 0 & 0 & 1 & 0 & -3 \\ 0 & 1 & 0 & 2 & -2 & 2 \\ 0 & 0 & 1 & -2 & 4 & -1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$
1. A basis of $$W$$ is $$\left\{ \left[\begin{array}{c} -3 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ -3 \end{array}\right] \right\}$$.
2. The dimension of $$W$$ is $$3$$.