Linear Algebra Examples

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 0.3 & 0.9 & 1.3 & 2.3 & 3.1 & 3.3 & 3.3 & 4.3 & 4.8 & 5.3 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

Perform the row operation $R_{2} = R_{2} - 0.3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 1.1.1

Perform the row operation $R_{2} = R_{2} - 0.3 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 0.3 - 0.3 \cdot 1 & 0.9 - 0.3 \cdot 2 & 1.3 - 0.3 \cdot 3 & 2.3 - 0.3 \cdot 4 & 3.1 - 0.3 \cdot 5 & 3.3 - 0.3 \cdot 6 & 3.3 - 0.3 \cdot 7 & 4.3 - 0.3 \cdot 8 & 4.8 - 0.3 \cdot 9 & 5.3 - 0.3 \cdot 10 \end{array}]$

Step 1.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 0 & 0.3 & 0.4 & 1.1 & 1.6 & 1.5 & 1.2 & 1.9 & 2.1 & 2.3 \end{array}]$

Step 1.2

Multiply each element of $R_{2}$ by $\frac{1}{0.3}$ to make the entry at $2, 2$ a $1$ .

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Step 1.2.1

Multiply each element of $R_{2}$ by $\frac{1}{0.3}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \frac{0}{0.3} & \frac{0.3}{0.3} & \frac{0.4}{0.3} & \frac{1.1}{0.3} & \frac{1.6}{0.3} & \frac{1.5}{0.3} & \frac{1.2}{0.3} & \frac{1.9}{0.3} & \frac{2.1}{0.3} & \frac{2.3}{0.3} \end{array}]$

Step 1.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 0 & 1 & 1.\overline{3} & 3.\overline{6} & 5.\overline{3} & 5 & 4 & 6.\overline{3} & 7 & 7.\overline{6} \end{array}]$

Step 1.3

Perform the row operation $R_{1} = R_{1} - 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 1.3.1

Perform the row operation $R_{1} = R_{1} - 2 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 3 - 2 \cdot 1.\overline{3} & 4 - 2 \cdot 3.\overline{6} & 5 - 2 \cdot 5.\overline{3} & 6 - 2 \cdot 5 & 7 - 2 \cdot 4 & 8 - 2 \cdot 6.\overline{3} & 9 - 2 \cdot 7 & 10 - 2 \cdot 7.\overline{6} \\ 0 & 1 & 1.\overline{3} & 3.\overline{6} & 5.\overline{3} & 5 & 4 & 6.\overline{3} & 7 & 7.\overline{6} \end{array}]$

Step 1.3.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0.\overline{3} & - 3.\overline{3} & - 5.\overline{6} & - 4 & - 1 & - 4.\overline{6} & - 5 & - 5.\overline{3} \\ 0 & 1 & 1.\overline{3} & 3.\overline{6} & 5.\overline{3} & 5 & 4 & 6.\overline{3} & 7 & 7.\overline{6} \end{array}]$

Step 2

The pivot positions are the locations with the leading $1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions: $a_{11}$ and $a_{22}$

Pivot Columns: $1$ and $2$