## V6 - Basis identification

#### Example 1 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^3$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^3$$

#### Example 2 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -3 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^3$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^3$$

#### Example 3 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 2 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 2 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 2 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 2 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 4 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ 3 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ -4 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ 3 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 0 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ 2 \\ -2 \\ 0 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ 3 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ -4 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ 3 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 0 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ 2 \\ -2 \\ 0 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ 3 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ -4 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ 3 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 0 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ 2 \\ -2 \\ 0 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ 3 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ -4 \\ -2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ 3 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 0 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ 2 \\ -2 \\ 0 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 5 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 0 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -2 \\ 4 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 0 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -2 \\ 4 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 0 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -2 \\ 4 \\ -1 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 0 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -2 \\ 4 \\ -1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 6 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -1 \\ 3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -1 \\ 3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -1 \\ 3 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -1 \\ 3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 7 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -2 \\ 1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -2 \\ 1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -2 \\ 1 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -2 \\ 1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 8 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -7 \\ 3 \\ -5 \\ -4 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -7 \\ 3 \\ -5 \\ -4 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -7 \\ 3 \\ -5 \\ -4 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -3 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -7 \\ 3 \\ -5 \\ -4 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^4$$

#### Example 9 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -2 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ 1 \\ -4 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -2 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ 1 \\ -4 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -2 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ 1 \\ -4 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -2 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ 1 \\ -4 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 10 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 2 \\ 1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 2 \\ 1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 2 \\ 1 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 3 \\ 2 \\ 1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 11 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ 2 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ 2 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ 2 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ 2 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 12 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 2 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 3 \\ 1 \\ 3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 2 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 3 \\ 1 \\ 3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 2 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 3 \\ 1 \\ 3 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ 2 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -2 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 3 \\ 1 \\ 3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 13 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 3 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 3 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 3 \\ -1 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 3 \\ -1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 14 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -4 \\ -5 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -5 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 39 \\ 18 \\ 30 \\ 21 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc} 2 & 3 & -5 & -5 & 0 \\ -4 & -5 & -3 & -4 & 39 \\ -5 & 0 & 0 & -1 & 18 \\ -2 & -3 & 0 & -5 & 30 \\ -3 & -1 & 0 & -3 & 21 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 0 & -3 \\ 0 & 1 & 0 & 0 & -3 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -4 \\ -5 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -5 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 39 \\ 18 \\ 30 \\ 21 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -4 \\ -5 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -5 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 39 \\ 18 \\ 30 \\ 21 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -4 \\ -5 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -5 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ 0 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 39 \\ 18 \\ 30 \\ 21 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^5$$

#### Example 15 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ -1 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ -1 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ -1 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ -1 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ -1 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ -1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 16 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 3 \\ 3 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} -12 \\ -11 \\ -8 \\ -6 \\ -13 \end{array}\right] , \left[\begin{array}{c} 12 \\ 8 \\ -4 \\ 0 \\ 16 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc} 3 & 0 & -3 & -12 & 12 \\ 3 & 1 & -2 & -11 & 8 \\ 3 & 4 & 1 & -8 & -4 \\ 2 & 2 & 0 & -6 & 0 \\ 3 & -1 & -4 & -13 & 16 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & -4 & 4 \\ 0 & 1 & 1 & 1 & -4 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 3 \\ 3 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} -12 \\ -11 \\ -8 \\ -6 \\ -13 \end{array}\right] , \left[\begin{array}{c} 12 \\ 8 \\ -4 \\ 0 \\ 16 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 3 \\ 3 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} -12 \\ -11 \\ -8 \\ -6 \\ -13 \end{array}\right] , \left[\begin{array}{c} 12 \\ 8 \\ -4 \\ 0 \\ 16 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 3 \\ 3 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 1 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} -12 \\ -11 \\ -8 \\ -6 \\ -13 \end{array}\right] , \left[\begin{array}{c} 12 \\ 8 \\ -4 \\ 0 \\ 16 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^5$$

#### Example 17 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \end{array}\right] \right\}$$both spans $$\mathbb{R}^3$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^3$$

#### Example 18 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ 6 \\ -3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ 6 \\ -3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ 6 \\ -3 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^3$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ 6 \\ -3 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^3$$

#### Example 19 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -5 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ -1 \\ 3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -5 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ -1 \\ 3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -5 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ -1 \\ 3 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -5 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ -1 \\ 3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 20 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 4 \\ -3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 4 \\ -3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 4 \\ -3 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 4 \\ -3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 21 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -5 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -5 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -5 \\ -1 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -5 \\ -1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 22 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -4 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -4 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -4 \end{array}\right] \right\}$$both spans $$\mathbb{R}^3$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ -4 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^3$$

#### Example 23 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -4 \\ -2 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ -1 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -3 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 2 \\ -3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -4 \\ -2 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ -1 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -3 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 2 \\ -3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -4 \\ -2 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ -1 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -3 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 2 \\ -3 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -4 \\ -2 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 3 \\ -1 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -3 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 2 \\ -3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 24 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -3 \\ -4 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -3 \\ -4 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -3 \\ -4 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -3 \\ -4 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 25 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 3 \\ 1 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -1 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 1 \\ -4 \\ -2 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 3 \\ 1 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -1 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 1 \\ -4 \\ -2 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 3 \\ 1 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -1 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 1 \\ -4 \\ -2 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 3 \\ 1 \\ -1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -1 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 1 \\ -4 \\ -2 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 26 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 4 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 4 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 4 \end{array}\right] \right\}$$both spans $$\mathbb{R}^3$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 4 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^3$$

#### Example 27 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 1 \\ 1 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -5 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 3 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 0 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 3 \\ -4 \\ 0 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 1 \\ 1 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -5 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 3 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 0 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 3 \\ -4 \\ 0 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 1 \\ 1 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -5 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 3 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 0 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 3 \\ -4 \\ 0 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 1 \\ 1 \\ 4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -5 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -1 \\ 3 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ 0 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 3 \\ -4 \\ 0 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 28 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -5 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ -4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -2 \\ -4 \\ -5 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -5 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ -4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -2 \\ -4 \\ -5 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -5 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ -4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -2 \\ -4 \\ -5 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 1 \\ 1 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -5 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ -4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 1 \\ -5 \\ -2 \\ -4 \\ -5 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 29 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -2 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ -4 \\ 0 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 6 \\ -3 \\ 3 \\ 7 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -7 \\ 6 \\ 3 \\ -6 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -2 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ -4 \\ 0 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 6 \\ -3 \\ 3 \\ 7 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -7 \\ 6 \\ 3 \\ -6 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -2 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ -4 \\ 0 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 6 \\ -3 \\ 3 \\ 7 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -7 \\ 6 \\ 3 \\ -6 \\ -1 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -2 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ -4 \\ 0 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 6 \\ -3 \\ 3 \\ 7 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -7 \\ 6 \\ 3 \\ -6 \\ -1 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^5$$

#### Example 30 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ 3 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 0 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ 4 \\ 3 \\ 1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ 3 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 0 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ 4 \\ 3 \\ 1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ 3 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 0 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ 4 \\ 3 \\ 1 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ 3 \\ 1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ 1 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 0 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ 4 \\ 3 \\ 1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 31 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ 3 \\ 1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ 3 \\ 1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ 3 \\ 1 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 3 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ 3 \\ 1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 32 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 4 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -15 \\ 9 \\ -10 \\ 3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 1 \\ 0 \\ 3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc} -5 & -1 & 3 & -15 & -2 \\ 4 & -3 & 1 & 9 & -3 \\ -1 & 4 & 2 & -10 & 1 \\ -4 & -3 & -4 & 3 & 0 \\ -4 & 2 & -2 & -6 & 3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & -1 & 0 \\ 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 4 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -15 \\ 9 \\ -10 \\ 3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 1 \\ 0 \\ 3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 4 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -15 \\ 9 \\ -10 \\ 3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 1 \\ 0 \\ 3 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 4 \\ -1 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ 4 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -15 \\ 9 \\ -10 \\ 3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 1 \\ 0 \\ 3 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^5$$

#### Example 33 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ -2 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ 3 \\ -2 \\ 0 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ -2 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ 3 \\ -2 \\ 0 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ -2 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ 3 \\ -2 \\ 0 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 4 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ -2 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ 3 \\ -2 \\ 0 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 34 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -5 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -5 \\ -3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -5 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -5 \\ -3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -5 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -5 \\ -3 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -5 \\ 3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -5 \\ -3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 35 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -1 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^3$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^3$$

#### Example 36 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 2 \\ -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -4 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ -3 \\ 4 \\ -1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 2 \\ -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -4 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ -3 \\ 4 \\ -1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 2 \\ -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -4 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ -3 \\ 4 \\ -1 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 2 \\ -2 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ -4 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ -3 \\ 4 \\ -1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 37 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \\ -3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \\ -3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \\ -3 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -5 \\ -4 \\ -3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 38 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^3$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^3$$

#### Example 39 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 2 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ 3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 2 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ 3 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 2 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ 3 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 2 \\ -3 \\ 3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ 3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^4$$

#### Example 40 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 7 \\ -4 \\ -4 \\ -13 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 9 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc} -3 & -5 & 7 & -2 \\ 2 & 3 & -4 & 1 \\ 2 & 3 & -4 & 1 \\ -5 & 4 & -13 & 9 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & -1 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 7 \\ -4 \\ -4 \\ -13 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 9 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 7 \\ -4 \\ -4 \\ -13 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 9 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 2 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 3 \\ 4 \end{array}\right] , \left[\begin{array}{c} 7 \\ -4 \\ -4 \\ -13 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 1 \\ 9 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^4$$

#### Example 41 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ 3 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -5 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 0 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ -5 \\ -5 \\ 4 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ 3 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -5 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 0 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ -5 \\ -5 \\ 4 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ 3 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -5 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 0 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ -5 \\ -5 \\ 4 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ 3 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 4 \\ -5 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ 0 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ -5 \\ -5 \\ 4 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 42 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 0 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -7 \\ 2 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \\ -4 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 0 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -7 \\ 2 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \\ -4 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 0 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -7 \\ 2 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \\ -4 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 0 \\ 0 \\ -4 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -7 \\ 2 \\ -3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \\ -4 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^4$$

#### Example 43 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 6 \\ -14 \\ 9 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 6 \\ -14 \\ 9 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 6 \\ -14 \\ 9 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^3$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 6 \\ -14 \\ 9 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^3$$

#### Example 44 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -2 \\ -3 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 11 \\ 0 \\ 11 \\ -1 \\ 5 \end{array}\right] , \left[\begin{array}{c} -41 \\ -1 \\ -40 \\ 2 \\ -19 \end{array}\right] , \left[\begin{array}{c} 83 \\ -2 \\ 85 \\ -11 \\ 37 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc} -5 & 3 & 11 & -41 & 83 \\ -2 & -1 & 0 & -1 & -2 \\ -3 & 4 & 11 & -40 & 85 \\ -3 & -2 & -1 & 2 & -11 \\ -3 & 1 & 5 & -19 & 37 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & -1 & 4 & -7 \\ 0 & 1 & 2 & -7 & 16 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -2 \\ -3 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 11 \\ 0 \\ 11 \\ -1 \\ 5 \end{array}\right] , \left[\begin{array}{c} -41 \\ -1 \\ -40 \\ 2 \\ -19 \end{array}\right] , \left[\begin{array}{c} 83 \\ -2 \\ 85 \\ -11 \\ 37 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -2 \\ -3 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 11 \\ 0 \\ 11 \\ -1 \\ 5 \end{array}\right] , \left[\begin{array}{c} -41 \\ -1 \\ -40 \\ 2 \\ -19 \end{array}\right] , \left[\begin{array}{c} 83 \\ -2 \\ 85 \\ -11 \\ 37 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -2 \\ -3 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 11 \\ 0 \\ 11 \\ -1 \\ 5 \end{array}\right] , \left[\begin{array}{c} -41 \\ -1 \\ -40 \\ 2 \\ -19 \end{array}\right] , \left[\begin{array}{c} 83 \\ -2 \\ 85 \\ -11 \\ 37 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^5$$

#### Example 45 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -3 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 2 \\ 9 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 4 \\ 12 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -3 \\ 2 \\ 2 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc} 2 & -1 & 5 & 6 & -2 \\ 0 & -2 & 2 & 4 & 0 \\ 3 & -3 & 9 & 12 & -3 \\ -2 & -3 & -1 & 2 & 2 \\ -2 & -3 & -1 & 2 & 2 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 2 & 2 & -1 \\ 0 & 1 & -1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -3 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 2 \\ 9 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 4 \\ 12 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -3 \\ 2 \\ 2 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -3 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 2 \\ 9 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 4 \\ 12 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -3 \\ 2 \\ 2 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^5$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -3 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 2 \\ 9 \\ -1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 4 \\ 12 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ -3 \\ 2 \\ 2 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^5$$

#### Example 46 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^3$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^3$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -1 \end{array}\right] \right\}$$both spans $$\mathbb{R}^3$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -1 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^3$$

#### Example 47 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -3 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 7 \\ -4 \\ -12 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ 1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc} -4 & -1 & -2 & 3 \\ -3 & -5 & 7 & -3 \\ 4 & 4 & -4 & 0 \\ -4 & 4 & -12 & 1 \end{array}\right] = \left[\begin{array}{cccc} 1 & 0 & 1 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -3 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 7 \\ -4 \\ -12 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ 1 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -3 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 7 \\ -4 \\ -12 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ 1 \end{array}\right] \right\}$$both spans $$\mathbb{R}^4$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -3 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ 7 \\ -4 \\ -12 \end{array}\right] , \left[\begin{array}{c} 3 \\ -3 \\ 0 \\ 1 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^4$$

#### Example 48 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -8 \\ -18 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 5 \\ -3 \\ 0 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -8 \\ -18 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 5 \\ -3 \\ 0 \end{array}\right] \right\}$$is not a basis of $$\mathbb{R}^4$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -8 \\ -18 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 5 \\ -3 \\ 0 \end{array}\right] \right\}$$either doesn't span $$\mathbb{R}^4$$ or is linearly dependent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -4 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ -5 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -8 \\ -18 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 0 \\ 5 \\ -3 \\ 0 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^4$$

#### Example 49 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 4 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ -1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 2 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 4 \\ -3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 4 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ -1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 2 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 4 \\ -3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

is equivalent to the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 4 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ -1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 2 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 4 \\ -3 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 4 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -4 \\ -1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ 2 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 4 \\ -3 \end{array}\right] \right\}$$ is a basis of $$\mathbb{R}^5$$

#### Example 50 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ -5 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -4 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ 0 \\ -3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.

1. Write an equivalent statement in terms of other vector properties.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc} 1 & 3 & 2 & -3 & -5 \\ 0 & 1 & 1 & -1 & -3 \\ -5 & -3 & -5 & -4 & -5 \\ -1 & -2 & -1 & 2 & 0 \\ -5 & 1 & 3 & -4 & -3 \end{array}\right] = \left[\begin{array}{ccccc} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & -2 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$
The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ -5 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -4 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ 0 \\ -3 \end{array}\right] \right\}$$is a basis of $$\mathbb{R}^5$$.
The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ -5 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -4 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ 0 \\ -3 \end{array}\right] \right\}$$both spans $$\mathbb{R}^5$$ and is linearly independent.
2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 0 \\ -5 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -4 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ 0 \\ -3 \end{array}\right] \right\}$$ is not a basis of $$\mathbb{R}^5$$