Examples

[\begin{array}{c} 2 & 0 \\ 2 & 4 \end{array}]

$[\begin{array}{c} 2 & 0 \\ 2 & 4 \end{array}]$

Step 1

Find the reduced row echelon form.

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Step 1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{2}

$\frac{1}{2}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

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Step 1.1.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{2}

$\frac{1}{2}$ to make the entry at

1, 1

$1, 1$ a

1

$1$ .

[\begin{array}{c} \frac{2}{2} & \frac{0}{2} \\ 2 & 4 \end{array}]

$[\begin{array}{c} \frac{2}{2} & \frac{0}{2} \\ 2 & 4 \end{array}]$

Step 1.1.2

Simplify

R_{1}

$R_{1}$ .

[\begin{array}{c} 1 & 0 \\ 2 & 4 \end{array}]

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

[\begin{array}{c} 1 & 0 \\ 2 & 4 \end{array}]

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

Step 1.2

Perform the row operation

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

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

2, 1

$2, 1$ a

0

$0$ .

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Step 1.2.1

Perform the row operation

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

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

2, 1

$2, 1$ a

0

$0$ .

[\begin{array}{c} 1 & 0 \\ 2 - 2 \cdot 1 & 4 - 2 \cdot 0 \end{array}]

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

Step 1.2.2

Simplify

R_{2}

$R_{2}$ .

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

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

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

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

Step 1.3

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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Step 1.3.1

Multiply each element of

R_{2}

$R_{2}$ by

\frac{1}{4}

$\frac{1}{4}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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

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

Step 1.3.2

Simplify

R_{2}

$R_{2}$ .

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

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

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

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

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

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

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}]

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

Step 3

The basis for

Row (A)

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

Row (A)

$Row (A)$ is the number of vectors in the basis.

Basis of

Row (A)

$Row (A)$ :

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

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

Dimension of

Row (A)

$Row (A)$ :

2

$2$

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