Finite Math Examples

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 0 & 0 4 & 0 0 & 0 0 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 1

Find the reduced row echelon form.

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Step 1.1

Swap

R_{2}

$R_{2}$ with

R_{1}

$R_{1}$ to put a nonzero entry at

1, 1

$1, 1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 4 & 0 0 & 0 0 & 0 0 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 1.2

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{4}

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

1, 1

$1, 1$ a

1

$1$ .

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Step 1.2.1

Multiply each element of

R_{1}

$R_{1}$ by

\frac{1}{4}

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

1, 1

$1, 1$ a

1

$1$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} \frac{4}{4} & \frac{0}{4} 0 & 0 0 & 0 0 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 1.2.2

Simplify

R_{1}

$R_{1}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 0 & 0 0 & 0 0 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 0 & 0 0 & 0 0 & 4 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 1.3

Swap

R_{4}

$R_{4}$ with

R_{2}

$R_{2}$ to put a nonzero entry at

2, 2

$2, 2$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 0 & 4 0 & 0 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 1.4

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 1.4.1

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{matrix} 1 & 0 \frac{0}{4} & \frac{4}{4} 0 & 0 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 1.4.2

Simplify

R_{2}

$R_{2}$ .

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 0 & 1 0 & 0 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 0 & 1 0 & 0 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

⎡ ⎢ ⎢ ⎢ ⎢ ⎣ \begin{matrix} 1 & 0 0 & 1 0 & 0 0 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎥ ⎥ ⎦

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

Step 2

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

c_{1} [\begin{matrix} 1 & 0 \end{matrix}] + c_{2} [\begin{matrix} 0 & 1 \end{matrix}] + c_{3} [\begin{matrix} 0 & 0 \end{matrix}] + c_{4} [\begin{matrix} 0 & 0 \end{matrix}]

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

Step 3

The basis for

Row (A)

$Row (A)$ 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{matrix} 1 & 0 \end{matrix}], [\begin{matrix} 0 & 1 \end{matrix}]}

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

Dimension of

Row (A)

$Row (A)$ :

2

$2$

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