Đại số Ví dụ

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

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

Bước 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.

Bước 2

Tìm dạng ma trận hàng bậc thang rút gọn.

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Bước 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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Bước 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} ⎤ ⎥ ⎥ ⎦

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

Bước 2.1.2

Rút gọn

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} ⎤ ⎥ ⎦

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

Bước 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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Bước 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} ⎤ ⎥ ⎦

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

Bước 2.2.2

Rút gọn

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} ⎤ ⎥ ⎦

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

Bước 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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Bước 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} ⎤ ⎥ ⎦

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

Bước 2.3.2

Rút gọn

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} ⎤ ⎥ ⎦

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

Bước 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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Bước 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} ⎤ ⎥ ⎥ ⎦

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

Bước 2.4.2

Rút gọn

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} ⎤ ⎥ ⎦

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

Bước 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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Bước 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} ⎤ ⎥ ⎥ ⎦

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

Bước 2.5.2

Rút gọn

$R_{3}$ .

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

Bước 2.6

Perform the row operation

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

$2, 3$ a

$0$ .

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Bước 2.6.1

Perform the row operation

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

$2, 3$ a

$0$ .

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

Bước 2.6.2

Rút gọn

$R_{2}$ .

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

Bước 2.7

Perform the row operation

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

$1, 3$ a

$0$ .

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Bước 2.7.1

Perform the row operation

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

$1, 3$ a

$0$ .

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

Bước 2.7.2

Rút gọn

$R_{1}$ .

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

Bước 2.8

Perform the row operation

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

$1, 2$ a

$0$ .

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Bước 2.8.1

Perform the row operation

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

$1, 2$ a

$0$ .

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

Bước 2.8.2

Rút gọn

$R_{1}$ .

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

Bước 3

The pivot positions are the locations with the leading

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

Pivot Positions:

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

$a_{33}$

Pivot Columns:

$1, 2,$ and

$3$

Bước 4

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

$0$

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