线性代数示例

$[\begin{array}{c} 1 & 1 & 0 & 5 \\ 1 & 1 & 1 & 3 \\ 3 & 3 & 2 & 11 \end{array}]$

解题步骤 1

求行简化阶梯形矩阵。

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解题步骤 1.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

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解题步骤 1.1.1

Perform the row operation $R_{2} = R_{2} - R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 0 & 5 \\ 1 - 1 & 1 - 1 & 1 - 0 & 3 - 5 \\ 3 & 3 & 2 & 11 \end{array}]$

解题步骤 1.1.2

化简 $R_{2}$ 。

$[\begin{array}{c} 1 & 1 & 0 & 5 \\ 0 & 0 & 1 & - 2 \\ 3 & 3 & 2 & 11 \end{array}]$

解题步骤 1.2

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

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解题步骤 1.2.1

Perform the row operation $R_{3} = R_{3} - 3 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 1 & 0 & 5 \\ 0 & 0 & 1 & - 2 \\ 3 - 3 \cdot 1 & 3 - 3 \cdot 1 & 2 - 3 \cdot 0 & 11 - 3 \cdot 5 \end{array}]$

解题步骤 1.2.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 1 & 0 & 5 \\ 0 & 0 & 1 & - 2 \\ 0 & 0 & 2 & - 4 \end{array}]$

解题步骤 1.3

Perform the row operation $R_{3} = R_{3} - 2 R_{2}$ to make the entry at $3, 3$ a $0$ .

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解题步骤 1.3.1

Perform the row operation $R_{3} = R_{3} - 2 R_{2}$ to make the entry at $3, 3$ a $0$ .

$[\begin{array}{c} 1 & 1 & 0 & 5 \\ 0 & 0 & 1 & - 2 \\ 0 - 2 \cdot 0 & 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & - 4 - 2 \cdot - 2 \end{array}]$

解题步骤 1.3.2

化简 $R_{3}$ 。

$[\begin{array}{c} 1 & 1 & 0 & 5 \\ 0 & 0 & 1 & - 2 \\ 0 & 0 & 0 & 0 \end{array}]$

解题步骤 2

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: $a_{11}$ and $a_{23}$

Pivot Columns: $1$ and $3$

线性代数 示例