## V5 - Linear independence

#### Example 1 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 5 \\ -3 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -2 \\ 5 \\ -8 \end{array}\right] , \left[\begin{array}{c} -6 \\ 4 \\ -3 \\ 3 \\ -4 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 5 \\ -3 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -2 \\ 5 \\ -8 \end{array}\right] , \left[\begin{array}{c} -6 \\ 4 \\ -3 \\ 3 \\ -4 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 2 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -6 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -3 \\ 0 \\ 1 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ -1 \\ -2 \\ 5 \\ -8 \end{array}\right] + x_{5} \left[\begin{array}{c} -6 \\ 4 \\ -3 \\ 3 \\ -4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 5 \\ -3 \\ 0 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -2 \\ 5 \\ -8 \end{array}\right] , \left[\begin{array}{c} -6 \\ 4 \\ -3 \\ 3 \\ -4 \end{array}\right] \right\}$$is linearly dependent.

#### Example 2 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -4 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -6 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -6 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 2 \\ -4 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -4 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -6 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -6 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 2 \\ -4 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 4 \\ 2 \\ -4 \\ -1 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -3 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -2 \\ -6 \\ -4 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 4 \\ -6 \\ -2 \\ -3 \end{array}\right] + x_{5} \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 2 \\ -4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ 2 \\ -4 \\ -1 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -6 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -6 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 2 \\ -4 \end{array}\right] \right\}$$is linearly independent.

#### Example 3 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ -2 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ -2 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ 3 \\ -3 \\ -5 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -5 \\ -1 \\ -3 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 1 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ -2 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 4 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 0 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -1 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -6 \\ 5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -10 \\ 6 \\ -7 \\ -14 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & -4 & -6 & 7 & 0 \\ 0 & 0 & 5 & -10 & 0 \\ -6 & -1 & -6 & 6 & 0 \\ 3 & -1 & 5 & -7 & 0 \\ -4 & -6 & 5 & -14 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 0 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -1 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -6 \\ 5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -10 \\ 6 \\ -7 \\ -14 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ 0 \\ -6 \\ 3 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 0 \\ -1 \\ -1 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ 5 \\ -6 \\ 5 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 7 \\ -10 \\ 6 \\ -7 \\ -14 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ 0 \\ -6 \\ 3 \\ -4 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ -1 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -6 \\ 5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -10 \\ 6 \\ -7 \\ -14 \end{array}\right] \right\}$$is linearly dependent.

#### Example 5 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ 0 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -6 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -1 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -5 \\ -2 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ -5 \\ 0 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -6 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ -4 \\ -1 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -5 \\ -2 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -4 \\ -5 \\ 0 \\ 4 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -1 \\ -6 \\ -4 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -4 \\ -1 \\ 0 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ 4 \\ 0 \\ -5 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 6 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -11 \\ 3 \\ 4 \\ 10 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 6 \\ 6 \\ -12 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -1 & 4 & -5 & -1 & 3 & 0 \\ -1 & 5 & 4 & -11 & 3 & 0 \\ -2 & -3 & -4 & 3 & 6 & 0 \\ -2 & -3 & -5 & 4 & 6 & 0 \\ 4 & 2 & -4 & 10 & -12 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 2 & -3 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -11 \\ 3 \\ 4 \\ 10 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 6 \\ 6 \\ -12 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -2 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 5 \\ -3 \\ -3 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ 4 \\ -4 \\ -5 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ -11 \\ 3 \\ 4 \\ 10 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ 3 \\ 6 \\ 6 \\ -12 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 5 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 4 \\ -4 \\ -5 \\ -4 \end{array}\right] , \left[\begin{array}{c} -1 \\ -11 \\ 3 \\ 4 \\ 10 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 6 \\ 6 \\ -12 \end{array}\right] \right\}$$is linearly dependent.

#### Example 7 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 5 \\ 1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 11 \\ 10 \\ -10 \\ 26 \\ -5 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -5 & -4 & -3 & 0 & 11 & 0 \\ -5 & -6 & -4 & 1 & 10 & 0 \\ -3 & -5 & 2 & 5 & -10 & 0 \\ -4 & 0 & -6 & 1 & 26 & 0 \\ -1 & -5 & -1 & -6 & -5 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -2 & 0 \\ 0 & 1 & 0 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & -3 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 5 \\ 1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 11 \\ 10 \\ -10 \\ 26 \\ -5 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -4 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -6 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 1 \\ 5 \\ 1 \\ -6 \end{array}\right] + x_{5} \left[\begin{array}{c} 11 \\ 10 \\ -10 \\ 26 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -5 \\ -3 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ 2 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 5 \\ 1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 11 \\ 10 \\ -10 \\ 26 \\ -5 \end{array}\right] \right\}$$is linearly dependent.

#### Example 8 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 2 \\ -3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 8 \\ 1 \\ 5 \\ -9 \\ 19 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} -3 & -3 & -1 & 8 & 0 \\ 2 & -2 & -1 & 1 & 0 \\ -3 & -1 & 0 & 5 & 0 \\ -4 & 5 & -3 & -9 & 0 \\ -5 & -6 & 2 & 19 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & -2 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 2 \\ -3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 8 \\ 1 \\ 5 \\ -9 \\ 19 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ 2 \\ -3 \\ -4 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ 5 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ -3 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 8 \\ 1 \\ 5 \\ -9 \\ 19 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 2 \\ -3 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 0 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 8 \\ 1 \\ 5 \\ -9 \\ 19 \end{array}\right] \right\}$$is linearly dependent.

#### Example 9 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 5 \\ -3 \\ -4 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -6 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 5 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -5 \\ -6 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 5 \\ -3 \\ -4 \\ 2 \\ 4 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \\ 3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -6 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 5 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -5 \\ -6 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 5 \\ -3 \\ -4 \\ 2 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -5 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -6 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 1 \\ 5 \\ 2 \\ -5 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -5 \\ -6 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 10 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ 11 \\ 1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ -6 \\ -5 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 4 & -5 & 4 & 5 & 2 & 0 \\ -5 & -6 & -5 & 11 & 5 & 0 \\ 5 & -5 & 5 & 1 & -6 & 0 \\ 1 & 0 & 1 & 5 & -5 & 0 \\ -1 & -5 & -1 & 2 & -3 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ 11 \\ 1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ -6 \\ -5 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 5 \\ 11 \\ 1 \\ 5 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 5 \\ -6 \\ -5 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -5 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ 5 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ 11 \\ 1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ -6 \\ -5 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

#### Example 11 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ -4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 9 \\ 11 \\ 8 \\ 8 \end{array}\right] , \left[\begin{array}{c} 5 \\ 0 \\ -4 \\ 1 \\ -3 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -6 & 4 & -4 & -2 & 5 & 0 \\ 0 & -5 & -2 & 9 & 0 & 0 \\ 3 & -4 & -2 & 11 & -4 & 0 \\ 3 & -5 & 0 & 8 & 1 & 0 \\ -2 & -6 & -2 & 8 & -3 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ -4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 9 \\ 11 \\ 8 \\ 8 \end{array}\right] , \left[\begin{array}{c} 5 \\ 0 \\ -4 \\ 1 \\ -3 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -6 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -5 \\ -4 \\ -5 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 0 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ 9 \\ 11 \\ 8 \\ 8 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ 0 \\ -4 \\ 1 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -5 \\ -4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ -2 \\ -2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 9 \\ 11 \\ 8 \\ 8 \end{array}\right] , \left[\begin{array}{c} 5 \\ 0 \\ -4 \\ 1 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

#### Example 12 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ -4 \\ -5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ -6 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ -4 \\ -5 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ -6 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ 4 \\ -4 \\ 5 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 5 \\ -4 \\ -5 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 2 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -6 \\ -2 \\ -5 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ -6 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 13 π

Consider the statement

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

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

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

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -6 \\ 1 \\ 1 \\ -1 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 3 \\ 3 \\ -5 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 3 \\ 1 \\ 4 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ -1 \\ 3 \\ 0 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

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

#### Example 14 π

Consider the statement

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

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

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

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ -3 \\ 2 \\ -2 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 5 \\ -6 \\ 4 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ -4 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ -1 \\ 3 \\ -1 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} -5 \\ 4 \\ -2 \\ 0 \\ -1 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

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

#### Example 15 π

Consider the statement

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

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

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

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ 5 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ 3 \\ -3 \\ 5 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 2 \\ 0 \\ -5 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 4 \\ 2 \\ -1 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ 3 \\ 5 \\ 0 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

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

#### Example 16 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -5 \\ 5 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -5 \\ -6 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 12 \\ -6 \\ 13 \\ 6 \\ -9 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -2 \\ -2 \\ 4 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 1 & -4 & -6 & 12 & 5 & 0 \\ -5 & 1 & -5 & -6 & -4 & 0 \\ 5 & 3 & -6 & 13 & -2 & 0 \\ 4 & 4 & -2 & 6 & -2 & 0 \\ -6 & 1 & -4 & -9 & 4 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 2 & 0 & 0 \\ 0 & 1 & 0 & -1 & 0 & 0 \\ 0 & 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -5 \\ 5 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -5 \\ -6 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 12 \\ -6 \\ 13 \\ 6 \\ -9 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -2 \\ -2 \\ 4 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ -5 \\ 5 \\ 4 \\ -6 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 4 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -5 \\ -6 \\ -2 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} 12 \\ -6 \\ 13 \\ 6 \\ -9 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ -4 \\ -2 \\ -2 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -5 \\ 5 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -5 \\ -6 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 12 \\ -6 \\ 13 \\ 6 \\ -9 \end{array}\right] , \left[\begin{array}{c} 5 \\ -4 \\ -2 \\ -2 \\ 4 \end{array}\right] \right\}$$is linearly dependent.

#### Example 17 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -2 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ 4 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 5 \\ 6 \\ -4 \\ 6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ 3 \\ -5 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -2 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ 4 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 5 \\ 6 \\ -4 \\ 6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ 3 \\ -5 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 4 \\ -2 \\ -6 \\ -1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -6 \\ 4 \\ -6 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ 5 \\ -2 \\ 4 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} 5 \\ 6 \\ -4 \\ 6 \\ 5 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ 3 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -2 \\ -6 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ 4 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ -2 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 5 \\ 6 \\ -4 \\ 6 \\ 5 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -3 \\ 3 \\ -5 \end{array}\right] \right\}$$is linearly dependent.

#### Example 18 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ -6 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 11 \\ 10 \\ -4 \\ -7 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} -2 & -3 & -2 & -4 & 0 \\ -3 & 3 & -2 & 11 & 0 \\ -6 & 1 & -5 & 10 & 0 \\ -6 & -6 & -5 & -4 & 0 \\ -1 & -5 & 0 & -7 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ -6 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 11 \\ 10 \\ -4 \\ -7 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -2 \\ -3 \\ -6 \\ -6 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ -6 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -5 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 11 \\ 10 \\ -4 \\ -7 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ -6 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 3 \\ 1 \\ -6 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -5 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 11 \\ 10 \\ -4 \\ -7 \end{array}\right] \right\}$$is linearly dependent.

#### Example 19 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 5 \\ -5 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ 1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 10 \\ -4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -9 \\ -20 \\ 13 \\ 7 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 1 \\ -1 \\ -5 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 1 & 2 & 3 & -9 & -1 & 0 \\ 5 & 5 & 10 & -20 & -4 & 0 \\ -5 & 1 & -4 & 13 & 1 & 0 \\ -3 & 5 & 2 & 7 & -1 & 0 \\ 5 & -6 & -1 & -1 & -5 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 5 \\ -5 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ 1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 10 \\ -4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -9 \\ -20 \\ 13 \\ 7 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 1 \\ -1 \\ -5 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ 5 \\ -5 \\ -3 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 5 \\ 1 \\ 5 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 10 \\ -4 \\ 2 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} -9 \\ -20 \\ 13 \\ 7 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} -1 \\ -4 \\ 1 \\ -1 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ 5 \\ -5 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 5 \\ 1 \\ 5 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 10 \\ -4 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} -9 \\ -20 \\ 13 \\ 7 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -4 \\ 1 \\ -1 \\ -5 \end{array}\right] \right\}$$is linearly dependent.

#### Example 20 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 0 \\ -6 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 9 \\ 0 \\ -16 \\ -6 \\ -2 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} 3 & 3 & -6 & 9 & 0 \\ 0 & 0 & -4 & 0 & 0 \\ -6 & -4 & 4 & -16 & 0 \\ -5 & 4 & -2 & -6 & 0 \\ -3 & 4 & 1 & -2 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 2 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 0 \\ -6 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 9 \\ 0 \\ -16 \\ -6 \\ -2 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 3 \\ 0 \\ -6 \\ -5 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 0 \\ -4 \\ 4 \\ 4 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -4 \\ 4 \\ -2 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 9 \\ 0 \\ -16 \\ -6 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 3 \\ 0 \\ -6 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -6 \\ -4 \\ 4 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 9 \\ 0 \\ -16 \\ -6 \\ -2 \end{array}\right] \right\}$$is linearly dependent.

#### Example 21 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -4 \\ -5 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ 2 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 9 \\ -23 \\ -6 \\ 5 \\ -14 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & -4 & -1 & 9 & 0 \\ -4 & 3 & 5 & -23 & 0 \\ -5 & -1 & 2 & -6 & 0 \\ -3 & -4 & 2 & 5 & 0 \\ -2 & 2 & 3 & -14 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & -3 & 0 \\ 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -4 \\ -5 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ 2 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 9 \\ -23 \\ -6 \\ 5 \\ -14 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -4 \\ -5 \\ -3 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 3 \\ -1 \\ -4 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 5 \\ 2 \\ 2 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 9 \\ -23 \\ -6 \\ 5 \\ -14 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -4 \\ -5 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 5 \\ 2 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 9 \\ -23 \\ -6 \\ 5 \\ -14 \end{array}\right] \right\}$$is linearly dependent.

#### Example 22 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ -2 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 0 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -10 \\ -8 \\ -12 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 0 & -4 & 2 & -2 & -2 & 0 \\ -2 & -6 & 3 & 3 & 3 & 0 \\ 0 & -4 & -6 & -1 & -10 & 0 \\ 3 & 1 & 0 & 5 & -8 & 0 \\ 2 & 0 & -6 & 2 & -12 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ -2 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 0 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -10 \\ -8 \\ -12 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 0 \\ -2 \\ 0 \\ 3 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ 1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 0 \\ -6 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ 3 \\ -1 \\ 5 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} -2 \\ 3 \\ -10 \\ -8 \\ -12 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 0 \\ -2 \\ 0 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ -6 \\ -4 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 0 \\ -6 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -1 \\ 5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 3 \\ -10 \\ -8 \\ -12 \end{array}\right] \right\}$$is linearly dependent.

#### Example 23 π

Consider the statement

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

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

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

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -5 \\ 1 \\ 5 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 5 \\ -1 \\ 2 \\ 3 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 2 \\ -6 \\ -1 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 0 \\ -5 \\ 2 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

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

#### Example 24 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 5 \\ -6 \\ 5 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 9 \\ -2 \\ -3 \\ -2 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 5 \\ -6 \\ 5 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 9 \\ -2 \\ -3 \\ -2 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 5 \\ -6 \\ 5 \\ 3 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 0 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 2 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ -1 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} 4 \\ 9 \\ -2 \\ -3 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 5 \\ -6 \\ 5 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ -6 \\ 2 \\ -5 \end{array}\right] , \left[\begin{array}{c} -3 \\ -4 \\ -5 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 9 \\ -2 \\ -3 \\ -2 \end{array}\right] \right\}$$is linearly dependent.

#### Example 25 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 2 \\ 1 \\ -5 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \\ -4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 3 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 2 \\ 1 \\ -5 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 4 \\ 4 \\ 4 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 4 \\ -5 \\ 4 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ 3 \\ 0 \\ -2 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 0 \\ 2 \\ 1 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 26 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 4 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ -3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -1 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -4 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 4 \\ -4 \\ 4 \\ 4 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ -3 \\ -6 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -1 \\ -2 \\ 4 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -4 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -2 \\ 4 \\ -4 \\ 4 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 4 \\ -2 \\ -3 \\ -6 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 4 \\ -1 \\ -2 \\ 4 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -4 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 27 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 15 \\ 13 \\ 1 \\ -11 \\ 9 \end{array}\right] , \left[\begin{array}{c} 26 \\ 13 \\ -5 \\ -8 \\ 9 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & -5 & 15 & 26 & 0 \\ -1 & -6 & 13 & 13 & 0 \\ 1 & -1 & 1 & -5 & 0 \\ 3 & 4 & -11 & -8 & 0 \\ -5 & -2 & 9 & 9 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & -2 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 15 \\ 13 \\ 1 \\ -11 \\ 9 \end{array}\right] , \left[\begin{array}{c} 26 \\ 13 \\ -5 \\ -8 \\ 9 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 3 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -6 \\ -1 \\ 4 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 15 \\ 13 \\ 1 \\ -11 \\ 9 \end{array}\right] + x_{4} \left[\begin{array}{c} 26 \\ 13 \\ -5 \\ -8 \\ 9 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ -6 \\ -1 \\ 4 \\ -2 \end{array}\right] , \left[\begin{array}{c} 15 \\ 13 \\ 1 \\ -11 \\ 9 \end{array}\right] , \left[\begin{array}{c} 26 \\ 13 \\ -5 \\ -8 \\ 9 \end{array}\right] \right\}$$is linearly dependent.

#### Example 28 π

Consider the statement

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

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

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

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -6 \\ 1 \\ -6 \\ 3 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ -4 \\ -3 \\ -6 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 0 \\ -5 \\ 1 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -5 \\ 2 \\ 4 \\ 1 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

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

#### Example 29 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -4 \\ -6 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -6 \\ 5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 3 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -16 \\ -5 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 5 \\ -2 \\ 5 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -4 \\ -6 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -6 \\ 5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 3 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -16 \\ -5 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 5 \\ -2 \\ 5 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -1 \\ -4 \\ -6 \\ 0 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -6 \\ 5 \\ -2 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ -4 \\ 3 \\ -5 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ -16 \\ -5 \\ 1 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} -2 \\ -1 \\ 5 \\ -2 \\ 5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ -4 \\ -6 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -6 \\ 5 \\ -2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -4 \\ 3 \\ -5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -4 \\ -16 \\ -5 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ 5 \\ -2 \\ 5 \end{array}\right] \right\}$$is linearly dependent.

#### Example 30 π

Consider the statement

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

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

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

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -2 \\ 2 \\ 0 \\ 2 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 0 \\ 3 \\ 3 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 1 \\ 3 \\ 3 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -6 \\ 3 \\ 2 \\ 4 \\ -5 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

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

#### Example 31 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -6 \\ -4 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ -2 \\ 3 \\ -4 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 4 \\ -6 \\ -4 \\ 2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ -2 \\ 3 \\ -4 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 4 \\ -6 \\ -4 \\ 2 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 1 \\ 3 \\ -3 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -3 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 0 \\ -2 \\ 3 \\ -4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 32 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -1 \\ 4 \\ -6 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 20 \\ -2 \\ -32 \\ 37 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -3 & -1 & -3 & -2 & 20 & 0 \\ -1 & 1 & 1 & -6 & -2 & 0 \\ 4 & 4 & 4 & 3 & -32 & 0 \\ -6 & -5 & -3 & 3 & 37 & 0 \\ 0 & -6 & 5 & 2 & -3 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 0 & -2 & 0 \\ 0 & 0 & 1 & 0 & -3 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -1 \\ 4 \\ -6 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 20 \\ -2 \\ -32 \\ 37 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ -1 \\ 4 \\ -6 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ -5 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ -3 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ 3 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} 20 \\ -2 \\ -32 \\ 37 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -1 \\ 4 \\ -6 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ 4 \\ -5 \\ -6 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -6 \\ 3 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 20 \\ -2 \\ -32 \\ 37 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

#### Example 33 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 5 \\ 5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -15 \\ -15 \\ 15 \\ 16 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} -2 & 4 & 2 & 1 & 0 \\ 5 & 3 & -5 & -15 & 0 \\ 5 & 0 & -5 & -15 & 0 \\ -4 & -4 & 4 & 15 & 0 \\ -5 & -4 & 5 & 16 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 5 \\ 5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -15 \\ -15 \\ 15 \\ 16 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -2 \\ 5 \\ 5 \\ -4 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -4 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ 4 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -15 \\ -15 \\ 15 \\ 16 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 5 \\ 5 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 4 \\ 3 \\ 0 \\ -4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 2 \\ -5 \\ -5 \\ 4 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -15 \\ -15 \\ 15 \\ 16 \end{array}\right] \right\}$$is linearly dependent.

#### Example 34 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -2 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -6 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 1 \\ -3 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ -3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ -2 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -6 \\ 0 \\ -5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 3 \\ 1 \\ -3 \\ -3 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -4 \\ 2 \\ -4 \\ -3 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ -2 \\ -2 \\ 1 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -3 \\ -6 \\ 0 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 3 \\ 1 \\ -3 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 35 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -6 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ -4 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 12 \\ 2 \\ -2 \\ 6 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -6 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ -4 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 12 \\ 2 \\ -2 \\ 6 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 4 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -6 \\ 4 \\ -2 \\ 3 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ -1 \\ -4 \\ -4 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} 6 \\ 12 \\ 2 \\ -2 \\ 6 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -6 \\ 1 \\ 4 \\ -4 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -6 \\ 4 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ -4 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 6 \\ 12 \\ 2 \\ -2 \\ 6 \end{array}\right] \right\}$$is linearly dependent.

#### Example 36 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ 14 \\ -4 \\ 0 \\ 4 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ 14 \\ -4 \\ 0 \\ 4 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ -4 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ 4 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 2 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 6 \\ 14 \\ -4 \\ 0 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -6 \\ 4 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ 14 \\ -4 \\ 0 \\ 4 \end{array}\right] \right\}$$is linearly dependent.

#### Example 37 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ -3 \\ 5 \\ 3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ -3 \\ 1 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 16 \\ 11 \\ -33 \\ -7 \\ -23 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} -6 & 0 & -6 & 16 & 0 \\ -3 & 0 & -3 & 11 & 0 \\ 5 & 4 & 1 & -33 & 0 \\ 3 & 3 & 0 & -7 & 0 \\ 5 & 3 & 2 & -23 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 1 & 0 & 0 \\ 0 & 1 & -1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ -3 \\ 5 \\ 3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ -3 \\ 1 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 16 \\ 11 \\ -33 \\ -7 \\ -23 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -6 \\ -3 \\ 5 \\ 3 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 3 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -3 \\ 1 \\ 0 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 16 \\ 11 \\ -33 \\ -7 \\ -23 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ -3 \\ 5 \\ 3 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 4 \\ 3 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ -3 \\ 1 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 16 \\ 11 \\ -33 \\ -7 \\ -23 \end{array}\right] \right\}$$is linearly dependent.

#### Example 38 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -4 \\ 5 \\ -6 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ 0 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -6 \\ 3 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -4 \\ 5 \\ -6 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ 0 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -4 \\ -5 \\ -1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -6 \\ 3 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -4 \\ 5 \\ -6 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 2 \\ -4 \\ 1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -6 \\ 0 \\ 2 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ -4 \\ -5 \\ -1 \\ 3 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -6 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 39 π

Consider the statement

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

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

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

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 3 \\ 2 \\ 1 \\ 5 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 5 \\ -5 \\ -2 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -2 \\ 1 \\ -4 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 5 \\ 4 \\ 5 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} -2 \\ 5 \\ -5 \\ 0 \\ -3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

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

#### Example 40 π

Consider the statement

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

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

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

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 5 \\ -6 \\ -4 \\ -2 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 4 \\ -1 \\ -3 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ 1 \\ -6 \\ -1 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 4 \\ -2 \\ -1 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

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

#### Example 41 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ 5 \\ 3 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ 0 \\ -4 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 3 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 5 \\ 4 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -1 \\ 5 \\ 3 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} -6 \\ 0 \\ -4 \\ 1 \\ 3 \end{array}\right] , \left[\begin{array}{c} 5 \\ 3 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 5 \\ 4 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -1 \\ 5 \\ 3 \\ 1 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ 0 \\ -4 \\ 1 \\ 3 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ 3 \\ -1 \\ -3 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ -5 \\ 1 \\ 5 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 42 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -13 \\ -10 \\ 8 \\ -2 \end{array}\right] \right\}$$is linearly independent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} 2 & -2 & -2 & 7 & 0 \\ -2 & -1 & -1 & -13 & 0 \\ 5 & -1 & -1 & -10 & 0 \\ 4 & -4 & -4 & 8 & 0 \\ -3 & -5 & -5 & -2 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -13 \\ -10 \\ 8 \\ -2 \end{array}\right] \right\}$$is linearly independent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 2 \\ -2 \\ 5 \\ 4 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} 7 \\ -13 \\ -10 \\ 8 \\ -2 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has no nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} 2 \\ -2 \\ 5 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} -2 \\ -1 \\ -1 \\ -4 \\ -5 \end{array}\right] , \left[\begin{array}{c} 7 \\ -13 \\ -10 \\ 8 \\ -2 \end{array}\right] \right\}$$is linearly dependent.

#### Example 43 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 4 \\ 2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ -6 \\ 2 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ -6 \\ 1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 5 \\ 5 \\ -2 \\ -2 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ 4 \\ 2 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ -6 \\ 2 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ -6 \\ 1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 5 \\ 0 \end{array}\right] , \left[\begin{array}{c} 5 \\ 5 \\ -2 \\ -2 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -2 \\ 4 \\ 2 \\ 0 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 1 \\ -6 \\ 2 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ -6 \\ 1 \\ 1 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 1 \\ 2 \\ 5 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} 5 \\ 5 \\ -2 \\ -2 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 44 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -2 \\ 5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 0 \\ -1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ 1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 15 \\ -9 \\ 0 \\ 3 \\ 12 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & -5 & -6 & 15 & 0 \\ -2 & 3 & 5 & -9 & 0 \\ 3 & 0 & 1 & 0 & 0 \\ -2 & -1 & 2 & 3 & 0 \\ 5 & -4 & 5 & 12 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & -3 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -2 \\ 5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 0 \\ -1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ 1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 15 \\ -9 \\ 0 \\ 3 \\ 12 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -2 \\ 5 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 3 \\ 0 \\ -1 \\ -4 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ 5 \\ 1 \\ 2 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 15 \\ -9 \\ 0 \\ 3 \\ 12 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -5 \\ -2 \\ 3 \\ -2 \\ 5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 3 \\ 0 \\ -1 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 5 \\ 1 \\ 2 \\ 5 \end{array}\right] , \left[\begin{array}{c} 15 \\ -9 \\ 0 \\ 3 \\ 12 \end{array}\right] \right\}$$is linearly dependent.

#### Example 45 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -5 \\ -4 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -6 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ -3 \\ 1 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -3 \\ -5 \\ -4 \\ -6 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 3 \\ -6 \\ 3 \\ -5 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ -3 \\ 1 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ -5 \\ -4 \\ -6 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 0 \\ 4 \\ 0 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 3 \\ -6 \\ 3 \\ -5 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ 0 \\ -1 \\ -3 \\ 1 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 46 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ 4 \\ -6 \\ 5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 0 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 3 \\ 3 \\ -4 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ 4 \\ -6 \\ 5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 0 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 3 \\ 3 \\ -4 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -6 \\ 4 \\ -6 \\ 5 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 5 \\ -1 \\ 0 \\ -1 \\ -6 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -3 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -1 \\ 3 \\ 3 \\ -4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -6 \\ 4 \\ -6 \\ 5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -1 \\ 0 \\ -1 \\ -6 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ -3 \\ -3 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ 3 \\ 3 \\ -4 \end{array}\right] \right\}$$is linearly dependent.

#### Example 47 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 4 \\ 5 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ -1 \\ 5 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -4 \\ 4 \\ 5 \\ -3 \\ -5 \end{array}\right] , \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 5 \\ 4 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ -1 \\ 5 \\ 0 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -4 \\ 4 \\ 5 \\ -3 \\ -5 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 3 \\ 4 \\ -4 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 5 \\ 4 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ 1 \\ -1 \\ 5 \\ 0 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 48 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -5 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ -5 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ 4 \\ 4 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -5 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ -5 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ 4 \\ 4 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -4 \\ 4 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ -2 \\ -5 \\ 2 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 5 \\ -6 \\ -5 \\ -4 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ 4 \\ 4 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

2. The set of vectors $$\left\{ \left[\begin{array}{c} -2 \\ -3 \\ 2 \\ -4 \\ 4 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ -5 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 5 \\ -6 \\ -5 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 1 \\ 4 \\ 4 \end{array}\right] \right\}$$is linearly independent.

#### Example 49 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -2 \\ -6 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ -4 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -4 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 4 \\ -4 \\ 3 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

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

1. The statement

The set of vectors $$\left\{ \left[\begin{array}{c} 1 \\ -2 \\ -6 \\ 2 \\ -4 \end{array}\right] , \left[\begin{array}{c} -6 \\ 3 \\ -4 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ -5 \\ -4 \\ 0 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ 4 \\ 4 \\ -4 \\ 3 \end{array}\right] \right\}$$is linearly dependent.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ -2 \\ -6 \\ 2 \\ -4 \end{array}\right] + x_{2} \left[\begin{array}{c} -6 \\ 3 \\ -4 \\ 2 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ -5 \\ -4 \\ 0 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ 4 \\ 4 \\ -4 \\ 3 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.

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

#### Example 50 π

Consider the statement

The set of vectors $$\left\{ \left[\begin{array}{c} 5 \\ -2 \\ 3 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ 5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -16 \\ 1 \\ -10 \\ 12 \\ -7 \end{array}\right] \right\}$$is linearly dependent.

1. Write an equivalent statement using a vector equation.
2. Explain why your statement is true or false.

$\operatorname{RREF} \left[\begin{array}{cccc|c} 5 & -1 & 0 & -16 & 0 \\ -2 & -2 & -3 & 1 & 0 \\ 3 & 0 & -1 & -10 & 0 \\ -2 & 1 & 5 & 12 & 0 \\ 3 & 5 & -3 & -7 & 0 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$
The set of vectors $$\left\{ \left[\begin{array}{c} 5 \\ -2 \\ 3 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ 5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -16 \\ 1 \\ -10 \\ 12 \\ -7 \end{array}\right] \right\}$$is linearly dependent.
The vector equation $$x_{1} \left[\begin{array}{c} 5 \\ -2 \\ 3 \\ -2 \\ 3 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \\ 5 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ 5 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} -16 \\ 1 \\ -10 \\ 12 \\ -7 \end{array}\right] = \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \end{array}\right]$$has (infinitely many) nontrivial solutions.
2. The set of vectors $$\left\{ \left[\begin{array}{c} 5 \\ -2 \\ 3 \\ -2 \\ 3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ 0 \\ 1 \\ 5 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ 5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -16 \\ 1 \\ -10 \\ 12 \\ -7 \end{array}\right] \right\}$$is linearly dependent.