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## V6 - Basis identification

#### Example 1 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 $$\mathbb{R}^5$$


text:
###### Example3.1.12

Virginia is designing a rectangular garden. The garden's length must be $$4$$ meters less than three times the width, and its perimeter must be $$40$$ meters. Find the garden's length and width.

Reminder: A rectangle's perimeter formula is $$P=2(L+W)\text{,}$$ where $$P$$ stands for perimeter, $$L$$ stands for length and $$W$$ stands for width.

Assume the rectangle's width is $$W$$ meters. We can then represent the length as $$3W-4$$ meters since we are told that it is $$4$$ meters less than three times the width. It's given that the perimeter is $$40$$ meters. Substituting those values into the formula, we have:

\begin{align*} P\amp=2(L+W)\\ 40\amp=2(3W-4+W)\\ 40\amp=2(4W-4)\amp\text{Like terms were combined.} \end{align*}

The next step to solve this equation is to remove the parentheses by distribution.

To check this result, we'll want to replace $$6$$ in the equation $$40=2(4W-4)\text{:}$$
To determine the length, recall that this was represented by $$3W-4\text{,}$$ which is:
Thus the rectangle's width is $$6$$ meters and the length is $$14$$ meters.

format: plain


text:
ISSN 1000-1239 CN 11-1777/TP

• Paper •

### A Fast Matching Algorithm and Conflict Detection for Packet Filter Rules

Tian Daxin1, Liu Yanheng1, Li Yongli2, and Tang Yi1

1. 1(College of Computer Science and Technology, Jilin University, Changchun 130012) 2(Department of Computer Science, Northeast Normal University, Changchun 130024)
• Online:2005-07-15

Abstract: The bottleneck of the packet filter method is produced due to a large number of filter rules to be checked. A fast matching algorithm BSLT (binary search in leafs of tries) is presented, which is based on trie construction and only stores the matching rules in the leaf nodes, and thus it consumes less memory space. The space complexity is O(NW) where N is the number of filter rules, W is the maximum number of bits specified in the destination or source fields. Binary search is used in finding the matching rule in the leaf nodes, which speeds up matching. The complexities of both searching time and matching time are O(W) and O(N) respectively. Experimental performance evaluations show that the throughput of BSLT is about 20% greater than the sequence matching algorithm. Another problem, the rule conflict, is proved and a conflict detection algorithm is given. Experiments prove that the algorithm can detect the conflict correctly.


text:
# Recovery of Structural Controllability for Control Systems

Abstract : Fundamental problems in control systems theory are controllability and observability, and designing control systems so that these properties are satisfied or approximated sufficiently. However, it is prudent to assume that an attacker will not only be able to subvert measurements but also control the system. Moreover, an advanced adversary with an understanding of the control system may seek to take over control of the entire system or parts thereof, or deny the legitimate operator this capability. The effectiveness of such attacks has been demonstrated in previous work. Indeed, these attacks cannot be ruled out given the likely existence of unknown vulnerabilities, increasing connectivity of nominally air-gapped systems and supply chain issues. The ability to rapidly recover control after an attack has been initiated and to detect an adversary’s presence is, therefore, critical. This paper focuses on the problem of structural controllability, which has recently attracted substantial attention through the equivalent problem of the power dominating set introduced in the context of electrical power network control. However, these problems are known to be $\mathcal{NP}$-hard with poor approximability. Given their relevance to many networks, especially power networks, this paper studies strategies for the efficient restoration of controllability following attacks and attacker-defender interactions in power-law networks.
Keywords :
Type de document :
Communication dans un congrès
Jonathan Butts; Sujeet Shenoi. 8th International Conference on Critical Infrastructure Protection (ICCIP), Mar 2014, Arlington, United States. Springer, IFIP Advances in Information and Communication Technology, AICT-441, pp.47-63, 2014, Critical Infrastructure Protection VIII. 〈10.1007/978-3-662-45355-1_4〉
Domaine :

Littérature citée [16 références]

https://hal.inria.fr/hal-01386753
Contributeur : Hal Ifip <>
Soumis le : lundi 24 octobre 2016 - 15:31:26
Dernière modification le : lundi 24 octobre 2016 - 15:40:28

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### Citation

Cristina Alcaraz, Stephen Wolthusen. Recovery of Structural Controllability for Control Systems. Jonathan Butts; Sujeet Shenoi. 8th International Conference on Critical Infrastructure Protection (ICCIP), Mar 2014, Arlington, United States. Springer, IFIP Advances in Information and Communication Technology, AICT-441, pp.47-63, 2014, Critical Infrastructure Protection VIII. 〈10.1007/978-3-662-45355-1_4〉. 〈hal-01386753〉

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