线性代数示例

$y = 0$ , $y = 2$ , $y = \sqrt{x}$

解题步骤 1

从等式两边同时减去 $\sqrt{x}$ 。

$y = 0, y = 2, y - \sqrt{x} = 0$

解题步骤 2

以矩阵形式书写方程组。

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

解题步骤 3

求行简化阶梯形矩阵。

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解题步骤 3.1

Swap $R_{3}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

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

解题步骤 3.2

Multiply each element of $R_{1}$ by $- 1$ to make the entry at $1, 1$ a $1$ .

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解题步骤 3.2.1

Multiply each element of $R_{1}$ by $- 1$ to make the entry at $1, 1$ a $1$ .

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

解题步骤 3.2.2

化简 $R_{1}$ 。

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

解题步骤 3.3

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

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解题步骤 3.3.1

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

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

解题步骤 3.3.2

化简 $R_{3}$ 。

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

解题步骤 3.4

Multiply each element of $R_{3}$ by $- \frac{1}{2}$ to make the entry at $3, 3$ a $1$ .

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解题步骤 3.4.1

Multiply each element of $R_{3}$ by $- \frac{1}{2}$ to make the entry at $3, 3$ a $1$ .

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

解题步骤 3.4.2

化简 $R_{3}$ 。

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

解题步骤 3.5

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

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解题步骤 3.5.1

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

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

解题步骤 3.5.2

化简 $R_{2}$ 。

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

解题步骤 3.6

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

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解题步骤 3.6.1

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

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

解题步骤 3.6.2

化简 $R_{1}$ 。

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

解题步骤 4

使用结果矩阵定义方程组的最终解。

$x = 0$

$y = 0$

$0 = 1$

解题步骤 5

因为 $0 \neq 1$ ，所以没有解。

无解

线性代数 示例

线性代数示例