Examples

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

Step 1

Find the reduced row echelon form.

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Step 1.1

Swap $R_{2}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

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

Step 1.2

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

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Step 1.2.1

Multiply each element of $R_{1}$ by $\frac{1}{4}$ to make the entry at $1, 1$ a $1$ .

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

Step 1.2.2

Simplify $R_{1}$ .

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

Step 1.3

Swap $R_{4}$ with $R_{2}$ to put a nonzero entry at $2, 2$ .

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

Step 1.4

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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Step 1.4.1

Multiply each element of $R_{2}$ by $\frac{1}{4}$ to make the entry at $2, 2$ a $1$ .

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

Step 1.4.2

Simplify $R_{2}$ .

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

Step 2

The row space of a matrix is the collection of all possible linear combinations of its row vectors.

$c_{1} [\begin{array}{c} 1 & 0 \end{array}] + c_{2} [\begin{array}{c} 0 & 1 \end{array}] + c_{3} [\begin{array}{c} 0 & 0 \end{array}] + c_{4} [\begin{array}{c} 0 & 0 \end{array}]$

Step 3

The basis for $Row (A)$ is the non-zero rows of a matrix in reduced row echelon form. The dimension of the basis for $Row (A)$ is the number of vectors in the basis.

Basis of $Row (A)$ : ${[\begin{array}{c} 1 & 0 \end{array}], [\begin{array}{c} 0 & 1 \end{array}]}$

Dimension of $Row (A)$ : $2$

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