Examples

A = [\begin{array}{c} 1 \\ 5 \\ 6 \end{array}]

$A = [\begin{array}{c} 1 \\ 5 \\ 6 \end{array}]$ ,

x = [\begin{array}{c} 1 \\ - 2 \\ 8 \end{array}]

$x = [\begin{array}{c} 1 \\ - 2 \\ 8 \end{array}]$

Step 1

C_{1} \cdot [\begin{array}{c} 1 \\ 5 \\ 6 \end{array}] = [\begin{array}{c} 1 \\ - 2 \\ 8 \end{array}]

$C_{1} \cdot [\begin{array}{c} 1 \\ 5 \\ 6 \end{array}] = [\begin{array}{c} 1 \\ - 2 \\ 8 \end{array}]$

Step 2

6 C_{1} = 8 C_{1} = 1 5 C_{1} = - 2

$6 C_{1} = 8 C_{1} = 1 5 C_{1} = - 2$

Step 3

Write the system of equations in matrix form.

[\begin{array}{c} 1 & 1 \\ 5 & - 2 \\ 6 & 8 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 5 & - 2 \\ 6 & 8 \end{array}]$

Step 4

Find the reduced row echelon form.

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Step 4.1

Perform the row operation

R_{2} = R_{2} - 5 R_{1}

$R_{2} = R_{2} - 5 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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Step 4.1.1

Perform the row operation

R_{2} = R_{2} - 5 R_{1}

$R_{2} = R_{2} - 5 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 1 \\ 5 - 5 \cdot 1 & - 2 - 5 \cdot 1 \\ 6 & 8 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 5 - 5 \cdot 1 & - 2 - 5 \cdot 1 \\ 6 & 8 \end{array}]$

Step 4.1.2

Simplify

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 6 & 8 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 6 & 8 \end{array}]$

[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 6 & 8 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 6 & 8 \end{array}]$

Step 4.2

Perform the row operation

R_{3} = R_{3} - 6 R_{1}

$R_{3} = R_{3} - 6 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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Step 4.2.1

Perform the row operation

R_{3} = R_{3} - 6 R_{1}

$R_{3} = R_{3} - 6 R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 6 - 6 \cdot 1 & 8 - 6 \cdot 1 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 6 - 6 \cdot 1 & 8 - 6 \cdot 1 \end{array}]$

Step 4.2.2

Simplify

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 0 & 2 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 0 & 2 \end{array}]$

[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 0 & 2 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & - 7 \\ 0 & 2 \end{array}]$

Step 4.3

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{7}

$- \frac{1}{7}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 4.3.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{7}

$- \frac{1}{7}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

[\begin{array}{c} 1 & 1 \\ - \frac{1}{7} \cdot 0 & - \frac{1}{7} \cdot - 7 \\ 0 & 2 \end{array}]

$[\begin{array}{c} 1 & 1 \\ - \frac{1}{7} \cdot 0 & - \frac{1}{7} \cdot - 7 \\ 0 & 2 \end{array}]$

Step 4.3.2

Simplify

R_{2}

$R_{2}$ .

[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 & 2 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 & 2 \end{array}]$

[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 & 2 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 & 2 \end{array}]$

Step 4.4

Perform the row operation

R_{3} = R_{3} - 2 R_{2}

$R_{3} = R_{3} - 2 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

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Step 4.4.1

Perform the row operation

R_{3} = R_{3} - 2 R_{2}

$R_{3} = R_{3} - 2 R_{2}$ to make the entry at

3, 2

$3, 2$ a

0

$0$ .

[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 \end{array}]$

Step 4.4.2

Simplify

R_{3}

$R_{3}$ .

[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

Step 4.5

Perform the row operation

R_{1} = R_{1} - R_{2}

$R_{1} = R_{1} - R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

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Step 4.5.1

Perform the row operation

R_{1} = R_{1} - R_{2}

$R_{1} = R_{1} - R_{2}$ to make the entry at

1, 2

$1, 2$ a

0

$0$ .

[\begin{array}{c} 1 - 0 & 1 - 1 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 - 0 & 1 - 1 \\ 0 & 1 \\ 0 & 0 \end{array}]$

Step 4.5.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]$

[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]$

[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]

$[\begin{array}{c} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}]$

Step 5

Use the result matrix to declare the final solutions to the system of equations.

C_{1} = 0

$C_{1} = 0$

0 = 1

$0 = 1$

Step 6

Since

0 \neq 1

$0 \neq 1$ , there are no solutions.

No solution

Step 7

There is not a transformation of the vector that exists because there was no unique solution to the system of equations. Since there is no linear transformation, the vector is not in the column space.

Not in the Column Space

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