Linear Algebra Examples

$[\begin{array}{c} 4 & 2 & 3 & 6 & - 7 \\ 1 & 0 & 1 & 0 & 1 \end{array}]$

Assign the matrix the name $A$ to simplify the descriptions throughout the problem.

$A = [\begin{array}{c} 4 & 2 & 3 & 6 & - 7 \\ 1 & 0 & 1 & 0 & 1 \end{array}]$

Find the reduced row echelon form of the matrix.

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Perform the row operation $R_{1} = \frac{1}{4} R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $1$ .

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Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = \frac{1}{4} R_{1}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} \frac{1}{4} R_{1} & \frac{1}{4} R_{1} & \frac{1}{4} R_{1} & \frac{1}{4} R_{1} & \frac{1}{4} R_{1} \\ 1 & 0 & 1 & 0 & 1 \end{array}]$

$R_{1} = \frac{1}{4} R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = \frac{1}{4} R_{1}$ .

$[\begin{array}{c} (\frac{1}{4}) \cdot (4) & (\frac{1}{4}) \cdot (2) & (\frac{1}{4}) \cdot (3) & (\frac{1}{4}) \cdot (6) & (\frac{1}{4}) \cdot (- 7) \\ 1 & 0 & 1 & 0 & 1 \end{array}]$

$R_{1} = \frac{1}{4} R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{3}{4} & \frac{3}{2} & - \frac{7}{4} \\ 1 & 0 & 1 & 0 & 1 \end{array}]$

Perform the row operation $R_{2} = - 1 \cdot R_{1} + R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - 1 \cdot R_{1} + R_{2}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{3}{4} & \frac{3}{2} & - \frac{7}{4} \\ - 1 \cdot R_{1} + R_{2} & - 1 \cdot R_{1} + R_{2} & - 1 \cdot R_{1} + R_{2} & - 1 \cdot R_{1} + R_{2} & - 1 \cdot R_{1} + R_{2} \end{array}]$

$R_{2} = - 1 \cdot R_{1} + R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 1 \cdot R_{1} + R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{3}{4} & \frac{3}{2} & - \frac{7}{4} \\ (- 1) \cdot (1) + 1 & (- 1) \cdot (\frac{1}{2}) + 0 & (- 1) \cdot (\frac{3}{4}) + 1 & (- 1) \cdot (\frac{3}{2}) + 0 & (- 1) \cdot (- \frac{7}{4}) + 1 \end{array}]$

$R_{2} = - 1 \cdot R_{1} + R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{3}{4} & \frac{3}{2} & - \frac{7}{4} \\ 0 & - \frac{1}{2} & \frac{1}{4} & - \frac{3}{2} & \frac{11}{4} \end{array}]$

Perform the row operation $R_{2} = - 2 \cdot R_{2}$ on $R_{2}$ (row $2$ ) in order to convert some elements in the row to $1$ .

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Replace $R_{2}$ (row $2$ ) with the row operation $R_{2} = - 2 \cdot R_{2}$ in order to convert some elements in the row to the desired value $1$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{3}{4} & \frac{3}{2} & - \frac{7}{4} \\ - 2 \cdot R_{2} & - 2 \cdot R_{2} & - 2 \cdot R_{2} & - 2 \cdot R_{2} & - 2 \cdot R_{2} \end{array}]$

$R_{2} = - 2 \cdot R_{2}$

Replace $R_{2}$ (row $2$ ) with the actual values of the elements for the row operation $R_{2} = - 2 \cdot R_{2}$ .

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{3}{4} & \frac{3}{2} & - \frac{7}{4} \\ (- 2) \cdot (0) & (- 2) \cdot (- \frac{1}{2}) & (- 2) \cdot (\frac{1}{4}) & (- 2) \cdot (- \frac{3}{2}) & (- 2) \cdot (\frac{11}{4}) \end{array}]$

$R_{2} = - 2 \cdot R_{2}$

Simplify $R_{2}$ (row $2$ ).

$[\begin{array}{c} 1 & \frac{1}{2} & \frac{3}{4} & \frac{3}{2} & - \frac{7}{4} \\ 0 & 1 & - \frac{1}{2} & 3 & - \frac{11}{2} \end{array}]$

Perform the row operation $R_{1} = - \frac{1}{2} R_{2} + R_{1}$ on $R_{1}$ (row $1$ ) in order to convert some elements in the row to $0$ .

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Replace $R_{1}$ (row $1$ ) with the row operation $R_{1} = - \frac{1}{2} R_{2} + R_{1}$ in order to convert some elements in the row to the desired value $0$ .

$[\begin{array}{c} - \frac{1}{2} R_{2} + R_{1} & - \frac{1}{2} R_{2} + R_{1} & - \frac{1}{2} R_{2} + R_{1} & - \frac{1}{2} R_{2} + R_{1} & - \frac{1}{2} R_{2} + R_{1} \\ 0 & 1 & - \frac{1}{2} & 3 & - \frac{11}{2} \end{array}]$

$R_{1} = - \frac{1}{2} R_{2} + R_{1}$

Replace $R_{1}$ (row $1$ ) with the actual values of the elements for the row operation $R_{1} = - \frac{1}{2} R_{2} + R_{1}$ .

$[\begin{array}{c} (- \frac{1}{2}) \cdot (0) + 1 & (- \frac{1}{2}) \cdot (1) + \frac{1}{2} & (- \frac{1}{2}) \cdot (- \frac{1}{2}) + \frac{3}{4} & (- \frac{1}{2}) \cdot (3) + \frac{3}{2} & (- \frac{1}{2}) \cdot (- \frac{11}{2}) - \frac{7}{4} \\ 0 & 1 & - \frac{1}{2} & 3 & - \frac{11}{2} \end{array}]$

$R_{1} = - \frac{1}{2} R_{2} + R_{1}$

Simplify $R_{1}$ (row $1$ ).

$[\begin{array}{c} 1 & 0 & 1 & 0 & 1 \\ 0 & 1 & - \frac{1}{2} & 3 & - \frac{11}{2} \end{array}]$

$D$ represents the set of the pivot columns in the row-reduced form of the matrix.

$D = {1, 2}$

Rank is the dimension of the column space of $A$ .

$r (A) = d i m (D)$

The rank of $A$ is $2$ .

$2$

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