## V2 - Linear combinations

#### Example 1 π

Consider the statement

The vector $$\left[\begin{array}{c} -4 \\ -8 \\ -9 \\ -7 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 1 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -2 \\ 4 \\ -2 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 2 & -1 & -5 & -3 & 4 & -4 \\ 1 & 2 & -1 & -2 & -2 & -8 \\ 2 & 1 & -4 & 3 & 4 & -9 \\ 2 & 1 & -2 & -2 & -2 & -7 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -\frac{134}{59} & -1 \\ 0 & 1 & 0 & 0 & -\frac{22}{59} & -3 \\ 0 & 0 & 1 & 0 & -\frac{112}{59} & 1 \\ 0 & 0 & 0 & 1 & \frac{26}{59} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} -4 \\ -8 \\ -9 \\ -7 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 1 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -2 \\ 4 \\ -2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 2 \\ 1 \\ 2 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ -2 \\ 3 \\ -2 \end{array}\right] + x_{5} \left[\begin{array}{c} 4 \\ -2 \\ 4 \\ -2 \end{array}\right] = \left[\begin{array}{c} -4 \\ -8 \\ -9 \\ -7 \end{array}\right]$$has no solutions.

2. $$\left[\begin{array}{c} -4 \\ -8 \\ -9 \\ -7 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 1 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} -1 \\ 2 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -4 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -2 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -2 \\ 4 \\ -2 \end{array}\right]$$.

#### Example 2 π

Consider the statement

The vector $$\left[\begin{array}{c} -9 \\ 4 \\ 9 \\ -3 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -9 \\ 4 \\ 9 \\ -3 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 2 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} -9 \\ 4 \\ 9 \\ -3 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 2 \end{array}\right]$$.

#### Example 3 π

Consider the statement

The vector $$\left[\begin{array}{c} -7 \\ 9 \\ -3 \\ -2 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -2 \\ -16 \\ -7 \end{array}\right] , \text{ and } \left[\begin{array}{c} 12 \\ 4 \\ 32 \\ 14 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{cccc|c} 3 & 0 & -6 & 12 & -7 \\ -2 & 2 & -2 & 4 & 9 \\ 2 & 4 & -16 & 32 & -3 \\ 2 & 1 & -7 & 14 & -2 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & -2 & 4 & 0 \\ 0 & 1 & -3 & 6 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} -7 \\ 9 \\ -3 \\ -2 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -2 \\ -16 \\ -7 \end{array}\right] , \text{ and } \left[\begin{array}{c} 12 \\ 4 \\ 32 \\ 14 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 3 \\ -2 \\ 2 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -6 \\ -2 \\ -16 \\ -7 \end{array}\right] + x_{4} \left[\begin{array}{c} 12 \\ 4 \\ 32 \\ 14 \end{array}\right] = \left[\begin{array}{c} -7 \\ 9 \\ -3 \\ -2 \end{array}\right]$$has no solutions.

2. $$\left[\begin{array}{c} -7 \\ 9 \\ -3 \\ -2 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ -2 \\ 2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -6 \\ -2 \\ -16 \\ -7 \end{array}\right] , \text{ and } \left[\begin{array}{c} 12 \\ 4 \\ 32 \\ 14 \end{array}\right]$$.

#### Example 4 π

Consider the statement

The vector $$\left[\begin{array}{c} 3 \\ -9 \\ 6 \\ 8 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 0 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -2 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -9 \\ 9 \\ -13 \\ 3 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 3 \\ -9 \\ 6 \\ 8 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 0 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -2 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -9 \\ 9 \\ -13 \\ 3 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 3 \\ -9 \\ 6 \\ 8 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 0 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -2 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -9 \\ 9 \\ -13 \\ 3 \end{array}\right]$$.

#### Example 5 π

Consider the statement

The vector $$\left[\begin{array}{c} 2 \\ -2 \\ -1 \\ -5 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 2 \\ -4 \\ -1 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -2 & 2 & 2 & 1 & 0 & 2 \\ -2 & 0 & 1 & 1 & 2 & -2 \\ -5 & 1 & 4 & 0 & -4 & -1 \\ -1 & -1 & -1 & 2 & -1 & -5 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & \frac{57}{4} & 2 \\ 0 & 1 & 0 & 0 & -\frac{43}{4} & 1 \\ 0 & 0 & 1 & 0 & \frac{39}{2} & 2 \\ 0 & 0 & 0 & 1 & 11 & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 2 \\ -2 \\ -1 \\ -5 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 2 \\ -4 \\ -1 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 2 \\ -2 \\ -1 \\ -5 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 0 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 2 \\ -4 \\ -1 \end{array}\right]$$.

#### Example 6 π

Consider the statement

The vector $$\left[\begin{array}{c} 8 \\ 6 \\ -2 \\ -10 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 0 \\ -4 \end{array}\right] , \text{ and } \left[\begin{array}{c} 6 \\ -2 \\ 0 \\ -8 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 8 \\ 6 \\ -2 \\ -10 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 0 \\ -4 \end{array}\right] , \text{ and } \left[\begin{array}{c} 6 \\ -2 \\ 0 \\ -8 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ 1 \\ -1 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 2 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ -1 \\ 0 \\ -4 \end{array}\right] + x_{4} \left[\begin{array}{c} 6 \\ -2 \\ 0 \\ -8 \end{array}\right] = \left[\begin{array}{c} 8 \\ 6 \\ -2 \\ -10 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 8 \\ 6 \\ -2 \\ -10 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 0 \\ -4 \end{array}\right] , \text{ and } \left[\begin{array}{c} 6 \\ -2 \\ 0 \\ -8 \end{array}\right]$$.

#### Example 7 π

Consider the statement

The vector $$\left[\begin{array}{c} 8 \\ -9 \\ -7 \\ 7 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -1 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 4 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 0 \\ -2 \\ -1 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 8 \\ -9 \\ -7 \\ 7 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -1 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 4 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 0 \\ -2 \\ -1 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ -1 \\ -5 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -1 \\ 4 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ 0 \\ -2 \\ -1 \end{array}\right] = \left[\begin{array}{c} 8 \\ -9 \\ -7 \\ 7 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 8 \\ -9 \\ -7 \\ 7 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -1 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 4 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 0 \\ -2 \\ -1 \end{array}\right]$$.

#### Example 8 π

Consider the statement

The vector $$\left[\begin{array}{c} 2 \\ -12 \\ 9 \\ -6 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ -2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 2 \\ -12 \\ 9 \\ -6 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ -2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ 1 \\ -1 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ -2 \end{array}\right] = \left[\begin{array}{c} 2 \\ -12 \\ 9 \\ -6 \end{array}\right]$$has no solutions.

2. $$\left[\begin{array}{c} 2 \\ -12 \\ 9 \\ -6 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -1 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ -3 \\ -5 \\ -2 \end{array}\right]$$.

#### Example 9 π

Consider the statement

The vector $$\left[\begin{array}{c} 13 \\ -7 \\ -11 \\ -4 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ -2 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{cccc|c} 4 & -1 & 4 & 1 & 13 \\ -1 & 0 & -3 & -1 & -7 \\ -5 & -2 & -4 & 3 & -11 \\ 1 & -1 & -3 & -2 & -4 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 13 \\ -7 \\ -11 \\ -4 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ -2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 4 \\ -1 \\ -5 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ -2 \end{array}\right] = \left[\begin{array}{c} 13 \\ -7 \\ -11 \\ -4 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 13 \\ -7 \\ -11 \\ -4 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ -2 \end{array}\right]$$.

#### Example 10 π

Consider the statement

The vector $$\left[\begin{array}{c} -8 \\ -10 \\ -3 \\ 1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -10 \\ 0 \\ -4 \\ 13 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -8 \\ -10 \\ -3 \\ 1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -10 \\ 0 \\ -4 \\ 13 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} -8 \\ -10 \\ -3 \\ 1 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -10 \\ 0 \\ -4 \\ 13 \end{array}\right]$$.

#### Example 11 π

Consider the statement

The vector $$\left[\begin{array}{c} 5 \\ -1 \\ 1 \\ -2 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -2 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 9 \\ -15 \\ 9 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 5 \\ -1 \\ 1 \\ -2 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -2 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 9 \\ -15 \\ 9 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 5 \\ -1 \\ 1 \\ -2 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -2 \\ 4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 9 \\ -15 \\ 9 \end{array}\right]$$.

#### Example 12 π

Consider the statement

The vector $$\left[\begin{array}{c} -8 \\ -7 \\ -7 \\ -6 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 8 \\ 13 \\ -6 \\ -5 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{cccc|c} -3 & 3 & 4 & 8 & -8 \\ -3 & 2 & -3 & 13 & -7 \\ 2 & -3 & -4 & -6 & -7 \\ 0 & -2 & 1 & -5 & -6 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -2 & 0 \\ 0 & 1 & 0 & 2 & 0 \\ 0 & 0 & 1 & -1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} -8 \\ -7 \\ -7 \\ -6 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 8 \\ 13 \\ -6 \\ -5 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 8 \\ 13 \\ -6 \\ -5 \end{array}\right] = \left[\begin{array}{c} -8 \\ -7 \\ -7 \\ -6 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -8 \\ -7 \\ -7 \\ -6 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -3 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -4 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 8 \\ 13 \\ -6 \\ -5 \end{array}\right]$$.

#### Example 13 π

Consider the statement

The vector $$\left[\begin{array}{c} 9 \\ -3 \\ -3 \\ -7 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ -2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 10 \\ 10 \\ -2 \\ -2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 9 \\ -3 \\ -3 \\ -7 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ -2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 10 \\ 10 \\ -2 \\ -2 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 9 \\ -3 \\ -3 \\ -7 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ -2 \\ 2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ -2 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 10 \\ 10 \\ -2 \\ -2 \end{array}\right]$$.

#### Example 14 π

Consider the statement

The vector $$\left[\begin{array}{c} 7 \\ 9 \\ -8 \\ -9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -10 \\ 7 \\ -2 \\ 5 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 7 \\ 9 \\ -8 \\ -9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -10 \\ 7 \\ -2 \\ 5 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -1 \\ -4 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} -10 \\ 7 \\ -2 \\ 5 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -2 \end{array}\right] = \left[\begin{array}{c} 7 \\ 9 \\ -8 \\ -9 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 7 \\ 9 \\ -8 \\ -9 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -10 \\ 7 \\ -2 \\ 5 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -3 \\ 3 \\ -2 \end{array}\right]$$.

#### Example 15 π

Consider the statement

The vector $$\left[\begin{array}{c} 9 \\ 4 \\ 7 \\ 3 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 3 \\ 15 \\ 3 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{cccc|c} 2 & 4 & 2 & -12 & 9 \\ 1 & 2 & -2 & 3 & 4 \\ -4 & -5 & -1 & 15 & 7 \\ 2 & -1 & -3 & 3 & 3 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & -3 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & -3 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 9 \\ 4 \\ 7 \\ 3 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 3 \\ 15 \\ 3 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ -2 \\ -1 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} -12 \\ 3 \\ 15 \\ 3 \end{array}\right] = \left[\begin{array}{c} 9 \\ 4 \\ 7 \\ 3 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 9 \\ 4 \\ 7 \\ 3 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 3 \\ 15 \\ 3 \end{array}\right]$$.

#### Example 16 π

Consider the statement

The vector $$\left[\begin{array}{c} 1 \\ -3 \\ 5 \\ -7 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 2 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 12 \\ 8 \\ -10 \\ -6 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccc|c} 3 & 3 & 12 & 1 \\ 2 & 2 & 8 & -3 \\ -3 & -2 & -10 & 5 \\ -1 & -2 & -6 & -7 \end{array}\right] = \left[\begin{array}{ccc|c} 1 & 0 & 2 & 0 \\ 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 1 \\ -3 \\ 5 \\ -7 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 2 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 12 \\ 8 \\ -10 \\ -6 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 3 \\ 2 \\ -3 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 2 \\ -2 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 12 \\ 8 \\ -10 \\ -6 \end{array}\right] = \left[\begin{array}{c} 1 \\ -3 \\ 5 \\ -7 \end{array}\right]$$has no solutions.

2. $$\left[\begin{array}{c} 1 \\ -3 \\ 5 \\ -7 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 2 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 12 \\ 8 \\ -10 \\ -6 \end{array}\right]$$.

#### Example 17 π

Consider the statement

The vector $$\left[\begin{array}{c} -2 \\ -6 \\ 9 \\ 0 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -3 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 2 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -3 & -1 & -5 & 2 & 0 & -2 \\ -3 & -1 & -1 & 1 & -3 & -6 \\ 4 & 1 & 0 & -4 & -4 & 9 \\ -1 & 1 & -1 & 0 & 2 & 0 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & \frac{24}{25} & 2 \\ 0 & 1 & 0 & 0 & \frac{72}{25} & 1 \\ 0 & 0 & 1 & 0 & -\frac{2}{25} & -1 \\ 0 & 0 & 0 & 1 & \frac{67}{25} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} -2 \\ -6 \\ 9 \\ 0 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -3 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ -3 \\ 4 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ -1 \\ 0 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 2 \end{array}\right] = \left[\begin{array}{c} -2 \\ -6 \\ 9 \\ 0 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -2 \\ -6 \\ 9 \\ 0 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ -3 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ 1 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ 0 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 1 \\ -4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 2 \end{array}\right]$$.

#### Example 18 π

Consider the statement

The vector $$\left[\begin{array}{c} -6 \\ 2 \\ -16 \\ -6 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -6 \\ 2 \\ -16 \\ -6 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ 0 \\ -2 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 3 \\ 2 \\ -4 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 2 \end{array}\right] = \left[\begin{array}{c} -6 \\ 2 \\ -16 \\ -6 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -6 \\ 2 \\ -16 \\ -6 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 0 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ -4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ 1 \\ 2 \\ 2 \end{array}\right]$$.

#### Example 19 π

Consider the statement

The vector $$\left[\begin{array}{c} 9 \\ 12 \\ 7 \\ -4 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 1 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -3 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 0 & -5 & -3 & 1 & 2 & 9 \\ -3 & -1 & -3 & -2 & 0 & 12 \\ -4 & -5 & -1 & 1 & 3 & 7 \\ 1 & 2 & 1 & -1 & -3 & -4 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -\frac{17}{20} & -1 \\ 0 & 1 & 0 & 0 & \frac{17}{20} & 0 \\ 0 & 0 & 1 & 0 & -\frac{6}{5} & -3 \\ 0 & 0 & 0 & 1 & \frac{53}{20} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 9 \\ 12 \\ 7 \\ -4 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 1 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -3 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ -1 \\ -5 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -3 \\ -1 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -2 \\ 1 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -3 \end{array}\right] = \left[\begin{array}{c} 9 \\ 12 \\ 7 \\ -4 \end{array}\right]$$has no solutions.

2. $$\left[\begin{array}{c} 9 \\ 12 \\ 7 \\ -4 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ -3 \\ -4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -5 \\ -1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -3 \\ -1 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 1 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 2 \\ 0 \\ 3 \\ -3 \end{array}\right]$$.

#### Example 20 π

Consider the statement

The vector $$\left[\begin{array}{c} 0 \\ 6 \\ 7 \\ -1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ 0 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ 2 \\ -1 \\ -1 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 0 \\ 6 \\ 7 \\ -1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ 0 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ 2 \\ -1 \\ -1 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 0 \\ 6 \\ 7 \\ -1 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ 0 \\ 4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ 2 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -4 \\ 2 \\ 4 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ 2 \\ -1 \\ -1 \end{array}\right]$$.

#### Example 21 π

Consider the statement

The vector $$\left[\begin{array}{c} 2 \\ 0 \\ 6 \\ 6 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ 0 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -3 \\ 0 \\ 2 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -1 & -4 & 0 & -1 & 1 & 2 \\ 0 & -3 & 2 & -3 & -3 & 0 \\ -3 & -3 & -4 & -1 & 0 & 6 \\ -3 & 0 & -3 & 2 & 2 & 6 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & \frac{47}{63} & -2 \\ 0 & 1 & 0 & 0 & -\frac{53}{63} & 0 \\ 0 & 0 & 1 & 0 & -\frac{1}{3} & 0 \\ 0 & 0 & 0 & 1 & \frac{34}{21} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 2 \\ 0 \\ 6 \\ 6 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ 0 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -3 \\ 0 \\ 2 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 2 \\ 0 \\ 6 \\ 6 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ 0 \\ -3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -3 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 2 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} -1 \\ -3 \\ -1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -3 \\ 0 \\ 2 \end{array}\right]$$.

#### Example 22 π

Consider the statement

The vector $$\left[\begin{array}{c} -8 \\ -6 \\ -6 \\ -1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -8 \\ -12 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ 8 \\ 8 \\ 0 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -8 \\ -6 \\ -6 \\ -1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -8 \\ -12 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ 8 \\ 8 \\ 0 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -2 \\ -3 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 2 \\ 3 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -2 \\ 1 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ -8 \\ -12 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} -5 \\ 8 \\ 8 \\ 0 \end{array}\right] = \left[\begin{array}{c} -8 \\ -6 \\ -6 \\ -1 \end{array}\right]$$has no solutions.

2. $$\left[\begin{array}{c} -8 \\ -6 \\ -6 \\ -1 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -2 \\ -3 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 4 \\ -2 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -8 \\ -12 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ 8 \\ 8 \\ 0 \end{array}\right]$$.

#### Example 23 π

Consider the statement

The vector $$\left[\begin{array}{c} 6 \\ 4 \\ -4 \\ -7 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 0 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -2 \\ 2 \\ -5 \\ 2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 6 \\ 4 \\ -4 \\ -7 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 0 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -2 \\ 2 \\ -5 \\ 2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 2 \\ -3 \\ -4 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -1 \\ 0 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -2 \\ 2 \\ -5 \\ 2 \end{array}\right] = \left[\begin{array}{c} 6 \\ 4 \\ -4 \\ -7 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 6 \\ 4 \\ -4 \\ -7 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ -3 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 0 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -2 \\ 2 \\ -5 \\ 2 \end{array}\right]$$.

#### Example 24 π

Consider the statement

The vector $$\left[\begin{array}{c} 0 \\ -1 \\ 4 \\ -1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 0 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 1 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 0 \\ -1 \\ 4 \\ -1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 0 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 1 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 0 \\ 0 \\ -2 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -1 \\ 2 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 0 \\ 0 \\ -1 \end{array}\right] + x_{5} \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 1 \end{array}\right] = \left[\begin{array}{c} 0 \\ -1 \\ 4 \\ -1 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 0 \\ -1 \\ 4 \\ -1 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ 0 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 2 \\ 0 \\ 1 \\ 1 \end{array}\right]$$.

#### Example 25 π

Consider the statement

The vector $$\left[\begin{array}{c} 4 \\ 7 \\ -8 \\ -3 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 7 \\ 10 \\ 0 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 4 \\ 7 \\ -8 \\ -3 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 7 \\ 10 \\ 0 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 4 \\ 7 \\ -8 \\ -3 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -3 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 1 \\ -5 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ -2 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 7 \\ 10 \\ 0 \end{array}\right]$$.

#### Example 26 π

Consider the statement

The vector $$\left[\begin{array}{c} -7 \\ -7 \\ -10 \\ 9 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -10 \\ 5 \\ -5 \\ 3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ -2 \\ -6 \\ 2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -7 \\ -7 \\ -10 \\ 9 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -10 \\ 5 \\ -5 \\ 3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ -2 \\ -6 \\ 2 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} -7 \\ -7 \\ -10 \\ 9 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -1 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 2 \\ 0 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -10 \\ 5 \\ -5 \\ 3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ -2 \\ -6 \\ 2 \end{array}\right]$$.

#### Example 27 π

Consider the statement

The vector $$\left[\begin{array}{c} -1 \\ 8 \\ -2 \\ 9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ 2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 1 \\ -2 \\ 2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -1 \\ 8 \\ -2 \\ 9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ 2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 1 \\ -2 \\ 2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ 1 \\ -2 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 1 \\ -1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -3 \\ 2 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 1 \\ -2 \\ 2 \end{array}\right] = \left[\begin{array}{c} -1 \\ 8 \\ -2 \\ 9 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -1 \\ 8 \\ -2 \\ 9 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ -2 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -3 \\ 2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 1 \\ -2 \\ 2 \end{array}\right]$$.

#### Example 28 π

Consider the statement

The vector $$\left[\begin{array}{c} 2 \\ -1 \\ 4 \\ 1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -3 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 2 \\ -1 \\ 4 \\ 1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -3 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -4 \\ -1 \\ -5 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 2 \\ 1 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -3 \end{array}\right] = \left[\begin{array}{c} 2 \\ -1 \\ 4 \\ 1 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 2 \\ -1 \\ 4 \\ 1 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ 1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -1 \\ -1 \\ -2 \\ -3 \end{array}\right]$$.

#### Example 29 π

Consider the statement

The vector $$\left[\begin{array}{c} -8 \\ 2 \\ 10 \\ -2 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ -1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -2 \\ -3 \\ -1 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -8 \\ 2 \\ 10 \\ -2 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ -1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -2 \\ -3 \\ -1 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 0 \\ 0 \\ -5 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -1 \\ -5 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -2 \\ -1 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ -2 \\ -3 \\ -1 \end{array}\right] = \left[\begin{array}{c} -8 \\ 2 \\ 10 \\ -2 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -8 \\ 2 \\ 10 \\ -2 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 0 \\ 0 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -2 \\ -1 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -2 \\ -3 \\ -1 \end{array}\right]$$.

#### Example 30 π

Consider the statement

The vector $$\left[\begin{array}{c} 7 \\ -2 \\ -16 \\ -2 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 4 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 2 \\ 3 \\ -3 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 7 \\ -2 \\ -16 \\ -2 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 4 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 2 \\ 3 \\ -3 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ 1 \\ 4 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ 1 \\ 2 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ 0 \\ 4 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 2 \\ 3 \\ -3 \end{array}\right] = \left[\begin{array}{c} 7 \\ -2 \\ -16 \\ -2 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 7 \\ -2 \\ -16 \\ -2 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 1 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 4 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 2 \\ 3 \\ -3 \end{array}\right]$$.

#### Example 31 π

Consider the statement

The vector $$\left[\begin{array}{c} 15 \\ 6 \\ 2 \\ -8 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ 1 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ 1 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 4 & -5 & -1 & 0 & -4 & 15 \\ 1 & 0 & -2 & 0 & -3 & 6 \\ -2 & -2 & -2 & -1 & 0 & 2 \\ -3 & -2 & 2 & 1 & 1 & -8 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -\frac{5}{13} & 2 \\ 0 & 1 & 0 & 0 & \frac{3}{13} & -1 \\ 0 & 0 & 1 & 0 & \frac{17}{13} & -2 \\ 0 & 0 & 0 & 1 & -\frac{30}{13} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 15 \\ 6 \\ 2 \\ -8 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ 1 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ 1 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 4 \\ 1 \\ -2 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -5 \\ 0 \\ -2 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 1 \end{array}\right] + x_{5} \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ 1 \end{array}\right] = \left[\begin{array}{c} 15 \\ 6 \\ 2 \\ -8 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 15 \\ 6 \\ 2 \\ -8 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ 1 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ 0 \\ -2 \\ -2 \end{array}\right] , \left[\begin{array}{c} -1 \\ -2 \\ -2 \\ 2 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ -3 \\ 0 \\ 1 \end{array}\right]$$.

#### Example 32 π

Consider the statement

The vector $$\left[\begin{array}{c} -4 \\ -4 \\ 4 \\ 1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 4 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ -1 \\ 1 \\ 0 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -4 \\ -4 \\ 4 \\ 1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 4 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ -1 \\ 1 \\ 0 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} -4 \\ -4 \\ 4 \\ 1 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 3 \\ 0 \\ 4 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ -1 \\ 1 \\ 0 \end{array}\right]$$.

#### Example 33 π

Consider the statement

The vector $$\left[\begin{array}{c} 7 \\ 5 \\ -3 \\ -9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -4 \\ -1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ -6 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 7 \\ 5 \\ -3 \\ -9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -4 \\ -1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ -6 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -4 \\ -1 \\ 1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -1 \\ -2 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ -6 \end{array}\right] = \left[\begin{array}{c} 7 \\ 5 \\ -3 \\ -9 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 7 \\ 5 \\ -3 \\ -9 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -4 \\ -1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ -2 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -4 \\ 0 \\ -4 \\ -6 \end{array}\right]$$.

#### Example 34 π

Consider the statement

The vector $$\left[\begin{array}{c} -1 \\ 0 \\ 3 \\ -6 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -12 \\ -9 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 3 \\ 15 \\ 9 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -1 \\ 0 \\ 3 \\ -6 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -12 \\ -9 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 3 \\ 15 \\ 9 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 4 \\ -1 \\ -5 \\ -3 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ -3 \\ -12 \\ -9 \end{array}\right] + x_{4} \left[\begin{array}{c} -12 \\ 3 \\ 15 \\ 9 \end{array}\right] = \left[\begin{array}{c} -1 \\ 0 \\ 3 \\ -6 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -1 \\ 0 \\ 3 \\ -6 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -1 \\ -5 \\ -3 \end{array}\right] , \left[\begin{array}{c} -4 \\ -1 \\ -2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 4 \\ -3 \\ -12 \\ -9 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 3 \\ 15 \\ 9 \end{array}\right]$$.

#### Example 35 π

Consider the statement

The vector $$\left[\begin{array}{c} 1 \\ 5 \\ -9 \\ -2 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 4 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -3 \\ -3 \\ 1 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 1 \\ 5 \\ -9 \\ -2 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 4 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -3 \\ -3 \\ 1 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 1 \\ 5 \\ -9 \\ -2 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 0 \\ 4 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ -3 \\ -3 \\ 1 \end{array}\right]$$.

#### Example 36 π

Consider the statement

The vector $$\left[\begin{array}{c} 8 \\ 4 \\ -3 \\ 1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -1 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 1 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ -2 \\ 1 \\ -3 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 8 \\ 4 \\ -3 \\ 1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -1 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 1 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ -2 \\ 1 \\ -3 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -1 \\ -1 \\ 2 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} -5 \\ -2 \\ 1 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} -5 \\ -2 \\ 1 \\ -3 \end{array}\right] = \left[\begin{array}{c} 8 \\ 4 \\ -3 \\ 1 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 8 \\ 4 \\ -3 \\ 1 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -1 \\ 2 \\ -2 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 3 \\ -3 \end{array}\right] , \left[\begin{array}{c} -5 \\ -2 \\ 1 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ -2 \\ 1 \\ -3 \end{array}\right]$$.

#### Example 37 π

Consider the statement

The vector $$\left[\begin{array}{c} 10 \\ 5 \\ 1 \\ -1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ 2 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 3 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ 2 \\ -2 \\ -2 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 10 \\ 5 \\ 1 \\ -1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ 2 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 3 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ 2 \\ -2 \\ -2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -1 \\ 2 \\ 2 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -4 \\ 0 \\ 3 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ 1 \\ 3 \\ -3 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ 2 \\ -2 \\ -2 \end{array}\right] = \left[\begin{array}{c} 10 \\ 5 \\ 1 \\ -1 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 10 \\ 5 \\ 1 \\ -1 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ 2 \\ 2 \\ 1 \end{array}\right] , \left[\begin{array}{c} -4 \\ 0 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 3 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ 2 \\ -2 \\ -2 \end{array}\right]$$.

#### Example 38 π

Consider the statement

The vector $$\left[\begin{array}{c} -2 \\ -3 \\ 4 \\ 9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 8 \\ 15 \\ 8 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -2 \\ -3 \\ 4 \\ 9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 8 \\ 15 \\ 8 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 3 \\ 1 \\ 1 \\ -1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ 1 \\ 3 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} -3 \\ 8 \\ 15 \\ 8 \end{array}\right] = \left[\begin{array}{c} -2 \\ -3 \\ 4 \\ 9 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -2 \\ -3 \\ 4 \\ 9 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 3 \\ 1 \\ 1 \\ -1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ -4 \\ -3 \end{array}\right] , \left[\begin{array}{c} 0 \\ 1 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 8 \\ 15 \\ 8 \end{array}\right]$$.

#### Example 39 π

Consider the statement

The vector $$\left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 4 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 2 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -1 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -2 & 3 & 1 & -2 & 0 & 2 \\ -3 & 1 & 0 & 1 & 1 & 0 \\ -3 & 4 & 0 & 2 & 1 & 0 \\ 2 & -1 & 2 & 1 & -1 & 4 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -\frac{4}{13} & 0 \\ 0 & 1 & 0 & 0 & -\frac{1}{26} & 0 \\ 0 & 0 & 1 & 0 & -\frac{7}{26} & 2 \\ 0 & 0 & 0 & 1 & \frac{3}{26} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 4 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 2 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -1 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 4 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 3 \\ 1 \\ 4 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ 0 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} -2 \\ 1 \\ 2 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ 1 \\ 1 \\ -1 \end{array}\right]$$.

#### Example 40 π

Consider the statement

The vector $$\left[\begin{array}{c} 2 \\ 7 \\ -7 \\ 5 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ 2 \\ 1 \\ -1 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 2 \\ 7 \\ -7 \\ 5 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ 2 \\ 1 \\ -1 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 2 \\ 7 \\ -7 \\ 5 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ 1 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 0 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ -3 \\ 1 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} -5 \\ 2 \\ 1 \\ -1 \end{array}\right]$$.

#### Example 41 π

Consider the statement

The vector $$\left[\begin{array}{c} -11 \\ 1 \\ 10 \\ 7 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 4 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 2 & 2 & 1 & 1 & -3 & -11 \\ 2 & -2 & -3 & -2 & 0 & 1 \\ -1 & -5 & 3 & 4 & 2 & 10 \\ -2 & -1 & 1 & -2 & -2 & 7 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -\frac{14}{11} & -3 \\ 0 & 1 & 0 & 0 & -\frac{2}{33} & -2 \\ 0 & 0 & 1 & 0 & -\frac{58}{33} & -1 \\ 0 & 0 & 0 & 1 & \frac{47}{33} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} -11 \\ 1 \\ 10 \\ 7 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 4 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 2 \\ 2 \\ -1 \\ -2 \end{array}\right] + x_{2} \left[\begin{array}{c} 2 \\ -2 \\ -5 \\ -1 \end{array}\right] + x_{3} \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -2 \\ 4 \\ -2 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \end{array}\right] = \left[\begin{array}{c} -11 \\ 1 \\ 10 \\ 7 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -11 \\ 1 \\ 10 \\ 7 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ 2 \\ -1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 2 \\ -2 \\ -5 \\ -1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -2 \\ 4 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ 0 \\ 2 \\ -2 \end{array}\right]$$.

#### Example 42 π

Consider the statement

The vector $$\left[\begin{array}{c} -4 \\ -10 \\ 3 \\ -2 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ 2 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 8 \\ -16 \\ 0 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -2 & -1 & -4 & -1 & -12 & -4 \\ 2 & -1 & 4 & -1 & 8 & -10 \\ -2 & -3 & -4 & -3 & -16 & 3 \\ 0 & 0 & 0 & 0 & 0 & -2 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 2 & 0 & 5 & 0 \\ 0 & 1 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} -4 \\ -10 \\ 3 \\ -2 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ 2 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 8 \\ -16 \\ 0 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -2 \\ 2 \\ -2 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -4 \\ 4 \\ -4 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} -12 \\ 8 \\ -16 \\ 0 \end{array}\right] = \left[\begin{array}{c} -4 \\ -10 \\ 3 \\ -2 \end{array}\right]$$has no solutions.

2. $$\left[\begin{array}{c} -4 \\ -10 \\ 3 \\ -2 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -2 \\ 2 \\ -2 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -4 \\ 4 \\ -4 \\ 0 \end{array}\right] , \left[\begin{array}{c} -1 \\ -1 \\ -3 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} -12 \\ 8 \\ -16 \\ 0 \end{array}\right]$$.

#### Example 43 π

Consider the statement

The vector $$\left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 1 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -5 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -1 \\ 3 \\ 2 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} -3 & -2 & -3 & 3 & 3 & 5 \\ 1 & 0 & 2 & -2 & -1 & -1 \\ 3 & 3 & 3 & -5 & 3 & -6 \\ 1 & 0 & -2 & 2 & 2 & -1 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & \frac{1}{2} & -1 \\ 0 & 1 & 0 & 0 & -\frac{9}{8} & -1 \\ 0 & 0 & 1 & 0 & -\frac{69}{16} & 0 \\ 0 & 0 & 0 & 1 & -\frac{57}{16} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -1 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 1 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -5 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -1 \\ 3 \\ 2 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -3 \\ 1 \\ 3 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} -2 \\ 0 \\ 3 \\ 0 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ 2 \\ 3 \\ -2 \end{array}\right] + x_{4} \left[\begin{array}{c} 3 \\ -2 \\ -5 \\ 2 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ -1 \\ 3 \\ 2 \end{array}\right] = \left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -1 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 5 \\ -1 \\ -6 \\ -1 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -3 \\ 1 \\ 3 \\ 1 \end{array}\right] , \left[\begin{array}{c} -2 \\ 0 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 2 \\ 3 \\ -2 \end{array}\right] , \left[\begin{array}{c} 3 \\ -2 \\ -5 \\ 2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -1 \\ 3 \\ 2 \end{array}\right]$$.

#### Example 44 π

Consider the statement

The vector $$\left[\begin{array}{c} 7 \\ -1 \\ 4 \\ -4 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -1 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 13 \\ 1 \\ 7 \\ 3 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} 7 \\ -1 \\ 4 \\ -4 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -1 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 13 \\ 1 \\ 7 \\ 3 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} 7 \\ -1 \\ 4 \\ -4 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ -1 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 3 \\ 2 \end{array}\right] , \left[\begin{array}{c} -5 \\ 1 \\ -1 \\ 1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 13 \\ 1 \\ 7 \\ 3 \end{array}\right]$$.

#### Example 45 π

Consider the statement

The vector $$\left[\begin{array}{c} -3 \\ 0 \\ 0 \\ -9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -14 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 6 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -1 \\ 16 \\ -5 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 1 & 0 & -3 & 1 & 3 & -3 \\ 1 & -1 & -1 & -1 & -1 & 0 \\ 4 & 1 & -14 & 6 & 16 & 0 \\ 1 & -2 & 1 & -3 & -5 & -9 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & -3 & 1 & 3 & 0 \\ 0 & 1 & -2 & 2 & 4 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} -3 \\ 0 \\ 0 \\ -9 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -14 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 6 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -1 \\ 16 \\ -5 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 1 \\ 1 \\ 4 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 0 \\ -1 \\ 1 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} -3 \\ -1 \\ -14 \\ 1 \end{array}\right] + x_{4} \left[\begin{array}{c} 1 \\ -1 \\ 6 \\ -3 \end{array}\right] + x_{5} \left[\begin{array}{c} 3 \\ -1 \\ 16 \\ -5 \end{array}\right] = \left[\begin{array}{c} -3 \\ 0 \\ 0 \\ -9 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -3 \\ 0 \\ 0 \\ -9 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 1 \\ 1 \\ 4 \\ 1 \end{array}\right] , \left[\begin{array}{c} 0 \\ -1 \\ 1 \\ -2 \end{array}\right] , \left[\begin{array}{c} -3 \\ -1 \\ -14 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -1 \\ 6 \\ -3 \end{array}\right] , \text{ and } \left[\begin{array}{c} 3 \\ -1 \\ 16 \\ -5 \end{array}\right]$$.

#### Example 46 π

Consider the statement

The vector $$\left[\begin{array}{c} 9 \\ 6 \\ -5 \\ 3 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ -7 \\ -11 \\ 7 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ -4 \\ -8 \\ -3 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 2 & 1 & 2 & 6 & -3 & 9 \\ -1 & -3 & 0 & -7 & -4 & 6 \\ -5 & -3 & 0 & -11 & -8 & -5 \\ 1 & 2 & 2 & 7 & -3 & 3 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 1 & 1 & 0 \\ 0 & 1 & 0 & 2 & 1 & 0 \\ 0 & 0 & 1 & 1 & -3 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} 9 \\ 6 \\ -5 \\ 3 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ -7 \\ -11 \\ 7 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ -4 \\ -8 \\ -3 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 2 \\ -1 \\ -5 \\ 1 \end{array}\right] + x_{2} \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 2 \end{array}\right] + x_{3} \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 2 \end{array}\right] + x_{4} \left[\begin{array}{c} 6 \\ -7 \\ -11 \\ 7 \end{array}\right] + x_{5} \left[\begin{array}{c} -3 \\ -4 \\ -8 \\ -3 \end{array}\right] = \left[\begin{array}{c} 9 \\ 6 \\ -5 \\ 3 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} 9 \\ 6 \\ -5 \\ 3 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} 2 \\ -1 \\ -5 \\ 1 \end{array}\right] , \left[\begin{array}{c} 1 \\ -3 \\ -3 \\ 2 \end{array}\right] , \left[\begin{array}{c} 2 \\ 0 \\ 0 \\ 2 \end{array}\right] , \left[\begin{array}{c} 6 \\ -7 \\ -11 \\ 7 \end{array}\right] , \text{ and } \left[\begin{array}{c} -3 \\ -4 \\ -8 \\ -3 \end{array}\right]$$.

#### Example 47 π

Consider the statement

The vector $$\left[\begin{array}{c} -12 \\ 1 \\ -1 \\ -3 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -2 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ 2 \\ 2 \\ 1 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{ccccc|c} 4 & 4 & 4 & 0 & 1 & -12 \\ -2 & 0 & 1 & 0 & 2 & 1 \\ 4 & -3 & 0 & 1 & 2 & -1 \\ 2 & 1 & 0 & 0 & 1 & -3 \end{array}\right] = \left[\begin{array}{ccccc|c} 1 & 0 & 0 & 0 & -\frac{11}{4} & -1 \\ 0 & 1 & 0 & 0 & \frac{13}{2} & -1 \\ 0 & 0 & 1 & 0 & -\frac{7}{2} & -1 \\ 0 & 0 & 0 & 1 & \frac{65}{2} & 0 \end{array}\right]$

1. The statement

The vector $$\left[\begin{array}{c} -12 \\ 1 \\ -1 \\ -3 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -2 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ 2 \\ 2 \\ 1 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} 4 \\ -2 \\ 4 \\ 2 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ 0 \\ -3 \\ 1 \end{array}\right] + x_{3} \left[\begin{array}{c} 4 \\ 1 \\ 0 \\ 0 \end{array}\right] + x_{4} \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] + x_{5} \left[\begin{array}{c} 1 \\ 2 \\ 2 \\ 1 \end{array}\right] = \left[\begin{array}{c} -12 \\ 1 \\ -1 \\ -3 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -12 \\ 1 \\ -1 \\ -3 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} 4 \\ -2 \\ 4 \\ 2 \end{array}\right] , \left[\begin{array}{c} 4 \\ 0 \\ -3 \\ 1 \end{array}\right] , \left[\begin{array}{c} 4 \\ 1 \\ 0 \\ 0 \end{array}\right] , \left[\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ 2 \\ 2 \\ 1 \end{array}\right]$$.

#### Example 48 π

Consider the statement

The vector $$\left[\begin{array}{c} -1 \\ -10 \\ 9 \\ -4 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 0 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ -1 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -1 \\ -10 \\ 9 \\ -4 \end{array}\right]$$is a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 0 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ -1 \end{array}\right]$$.

is equivalent to the statement

The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -3 \\ 3 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} -3 \\ 0 \\ 0 \\ -2 \end{array}\right] + x_{3} \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ -1 \end{array}\right] = \left[\begin{array}{c} -1 \\ -10 \\ 9 \\ -4 \end{array}\right]$$has a solution.

2. $$\left[\begin{array}{c} -1 \\ -10 \\ 9 \\ -4 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -3 \\ 3 \\ 0 \end{array}\right] , \left[\begin{array}{c} -3 \\ 0 \\ 0 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 0 \\ -3 \\ -1 \\ -1 \end{array}\right]$$.

#### Example 49 π

Consider the statement

The vector $$\left[\begin{array}{c} -9 \\ 4 \\ 0 \\ 1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -2 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -3 \\ 0 \\ -3 \end{array}\right]$$.

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

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

1. The statement

The vector $$\left[\begin{array}{c} -9 \\ 4 \\ 0 \\ 1 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -2 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -3 \\ 0 \\ -3 \end{array}\right]$$.

is equivalent to the statement

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

2. $$\left[\begin{array}{c} -9 \\ 4 \\ 0 \\ 1 \end{array}\right]$$ is not a linear combination of the vectors $$\left[\begin{array}{c} -1 \\ -2 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} -1 \\ 1 \\ -3 \\ -1 \end{array}\right] , \left[\begin{array}{c} 2 \\ 2 \\ 3 \\ -2 \end{array}\right] , \text{ and } \left[\begin{array}{c} 1 \\ -3 \\ 0 \\ -3 \end{array}\right]$$.

#### Example 50 π

Consider the statement

The vector $$\left[\begin{array}{c} -20 \\ 3 \\ -5 \\ 10 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 0 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 1 \\ 4 \\ -3 \end{array}\right]$$.

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

$\operatorname{RREF} \left[\begin{array}{cccc|c} -5 & 4 & 3 & 4 & -20 \\ -1 & -1 & -1 & 1 & 3 \\ 1 & 2 & 0 & 4 & -5 \\ 0 & -3 & -1 & -3 & 10 \end{array}\right] = \left[\begin{array}{cccc|c} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & -3 \\ 0 & 0 & 1 & 0 & -1 \\ 0 & 0 & 0 & 1 & 0 \end{array}\right]$
The vector $$\left[\begin{array}{c} -20 \\ 3 \\ -5 \\ 10 \end{array}\right]$$is not a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 0 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 1 \\ 4 \\ -3 \end{array}\right]$$.
The vector equation $$x_{1} \left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 0 \end{array}\right] + x_{2} \left[\begin{array}{c} 4 \\ -1 \\ 2 \\ -3 \end{array}\right] + x_{3} \left[\begin{array}{c} 3 \\ -1 \\ 0 \\ -1 \end{array}\right] + x_{4} \left[\begin{array}{c} 4 \\ 1 \\ 4 \\ -3 \end{array}\right] = \left[\begin{array}{c} -20 \\ 3 \\ -5 \\ 10 \end{array}\right]$$has no solutions.
2. $$\left[\begin{array}{c} -20 \\ 3 \\ -5 \\ 10 \end{array}\right]$$ is a linear combination of the vectors $$\left[\begin{array}{c} -5 \\ -1 \\ 1 \\ 0 \end{array}\right] , \left[\begin{array}{c} 4 \\ -1 \\ 2 \\ -3 \end{array}\right] , \left[\begin{array}{c} 3 \\ -1 \\ 0 \\ -1 \end{array}\right] , \text{ and } \left[\begin{array}{c} 4 \\ 1 \\ 4 \\ -3 \end{array}\right]$$.