الأمثلة

⎡ ⎢ ⎣ \begin{matrix} 2 & 4 & 6 8 & 10 & 12 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 2 & 4 & 6 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

خطوة 1

Nullity is the dimension of the null space, which is the same as the number of free variables in the system after row reducing. The free variables are the columns without pivot positions.

خطوة 2

أوجِد الصيغة الدرجية المختزلة صفيًا.

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خطوة 2.1

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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خطوة 2.1.1

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{matrix} \frac{2}{2} & \frac{4}{2} & \frac{6}{2} 8 & 10 & 12 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} \frac{2}{2} & \frac{4}{2} & \frac{6}{2} \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

خطوة 2.1.2

بسّط

R_{1}

$R_{1}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 8 & 10 & 12 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 8 & 10 & 12 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ 8 & 10 & 12 \\ 1 & 2 & 0 \end{array}]$

خطوة 2.2

Perform the row operation

R_{2} = R_{2} - 8 R_{1}

$R_{2} = R_{2} - 8 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

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خطوة 2.2.1

Perform the row operation

R_{2} = R_{2} - 8 R_{1}

$R_{2} = R_{2} - 8 R_{1}$ to make the entry at

2, 1

$2, 1$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 8 - 8 \cdot 1 & 10 - 8 \cdot 2 & 12 - 8 \cdot 3 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ 8 - 8 \cdot 1 & 10 - 8 \cdot 2 & 12 - 8 \cdot 3 \\ 1 & 2 & 0 \end{array}]$

خطوة 2.2.2

بسّط

R_{2}

$R_{2}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & - 6 & - 12 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 & 2 & 0 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & - 6 & - 12 1 & 2 & 0 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 & 2 & 0 \end{array}]$

خطوة 2.3

Perform the row operation

R_{3} = R_{3} - R_{1}

$R_{3} = R_{3} - R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

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خطوة 2.3.1

Perform the row operation

R_{3} = R_{3} - R_{1}

$R_{3} = R_{3} - R_{1}$ to make the entry at

3, 1

$3, 1$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & - 6 & - 12 1 - 1 & 2 - 2 & 0 - 3 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 1 - 1 & 2 - 2 & 0 - 3 \end{array}]$

خطوة 2.3.2

بسّط

R_{3}

$R_{3}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & - 6 & - 12 0 & 0 & - 3 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 0 & 0 & - 3 \end{array}]$

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & - 6 & - 12 0 & 0 & - 3 \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ 0 & - 6 & - 12 \\ 0 & 0 & - 3 \end{array}]$

خطوة 2.4

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{6}

$- \frac{1}{6}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

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خطوة 2.4.1

Multiply each element of

R_{2}

$R_{2}$ by

- \frac{1}{6}

$- \frac{1}{6}$ to make the entry at

2, 2

$2, 2$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 12 0 & 0 & - 3 \end{matrix} ⎤ ⎥ ⎥ ⎦

$[\begin{array}{c} 1 & 2 & 3 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot - 12 \\ 0 & 0 & - 3 \end{array}]$

خطوة 2.4.2

بسّط

R_{2}

$R_{2}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 2 0 & 0 & - 3 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 2 0 & 0 & - 3 \end{matrix} ⎤ ⎥ ⎦

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

خطوة 2.5

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{3}

$- \frac{1}{3}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

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خطوة 2.5.1

Multiply each element of

R_{3}

$R_{3}$ by

- \frac{1}{3}

$- \frac{1}{3}$ to make the entry at

3, 3

$3, 3$ a

1

$1$ .

⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 2 - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot 0 & - \frac{1}{3} \cdot - 3 \end{matrix} ⎤ ⎥ ⎥ ⎦

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

خطوة 2.5.2

بسّط

R_{3}

$R_{3}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 2 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 2 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

خطوة 2.6

Perform the row operation

R_{2} = R_{2} - 2 R_{3}

$R_{2} = R_{2} - 2 R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

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خطوة 2.6.1

Perform the row operation

R_{2} = R_{2} - 2 R_{3}

$R_{2} = R_{2} - 2 R_{3}$ to make the entry at

2, 3

$2, 3$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 - 2 \cdot 0 & 1 - 2 \cdot 0 & 2 - 2 \cdot 1 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

خطوة 2.6.2

بسّط

R_{2}

$R_{2}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 3 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

خطوة 2.7

Perform the row operation

R_{1} = R_{1} - 3 R_{3}

$R_{1} = R_{1} - 3 R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

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خطوة 2.7.1

Perform the row operation

R_{1} = R_{1} - 3 R_{3}

$R_{1} = R_{1} - 3 R_{3}$ to make the entry at

1, 3

$1, 3$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 - 3 \cdot 0 & 2 - 3 \cdot 0 & 3 - 3 \cdot 1 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

خطوة 2.7.2

بسّط

R_{1}

$R_{1}$ .

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

⎡ ⎢ ⎣ \begin{matrix} 1 & 2 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

خطوة 2.8

Perform the row operation

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

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

1, 2

$1, 2$ a

0

$0$ .

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خطوة 2.8.1

Perform the row operation

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

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

1, 2

$1, 2$ a

0

$0$ .

⎡ ⎢ ⎣ \begin{matrix} 1 - 2 \cdot 0 & 2 - 2 \cdot 1 & 0 - 2 \cdot 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦

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

خطوة 2.8.2

بسّط

R_{1}

$R_{1}$ .

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

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

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

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

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

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

خطوة 3

The pivot positions are the locations with the leading

1

$1$ in each row. The pivot columns are the columns that have a pivot position.

Pivot Positions:

a_{11}, a_{22},

$a_{11}, a_{22},$ and

a_{33}

$a_{33}$

Pivot Columns:

1, 2,

$1, 2,$ and

3

$3$

خطوة 4

The nullity is the number of columns without a pivot position in the row reduced matrix.

0

$0$

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