完成《线性代数》克莱姆法则
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title: "《线性代数》克莱姆法则"
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date: 2023-08-11T21:07:47+08:00
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---
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## 克莱姆法则
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克莱姆法则并非计算线性方程组的最好方法(高斯消元法),但能够加深对线性方程组的理解
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对于一个线性方程组
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$$
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\begin{cases}
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3x+2y &=-4 \\\\
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-1x+2y &=-2
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\end{cases}
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$$
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可以将其看作对向量的一个已知的矩阵变换,且结果已知
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$$
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\begin{bmatrix}
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3 & 2 \\\\
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-1 & 2
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\end{bmatrix}
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\begin{bmatrix}
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x \\\\
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y
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\end{bmatrix} =
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\begin{bmatrix}
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-4 \\\\
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-2
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\end{bmatrix}
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$$
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![](../../images/%E6%95%B0%E5%AD%A6/%E3%80%8A%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E3%80%8B%E5%85%8B%E8%8E%B1%E5%A7%86%E6%B3%95%E5%88%99/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84%E8%BD%AC%E5%8C%96.gif)
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$$
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x\begin{bmatrix}
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3 \\\\
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-1
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\end{bmatrix} +
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y\begin{bmatrix}
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2 \\\\
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2
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\end{bmatrix} =
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\begin{bmatrix}
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-4 \\\\
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-2
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\end{bmatrix}
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$$
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在计算时不能将点乘的结果视为x或y的坐标,因为点乘会随着线性变换而改变结果甚至正负性,但对于不改变点积的正交变换(基向量在变换后依然保持单位长度且相互垂直)则可以使用
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![](../../images/%E6%95%B0%E5%AD%A6/%E3%80%8A%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E3%80%8B%E5%85%8B%E8%8E%B1%E5%A7%86%E6%B3%95%E5%88%99/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5%E4%B8%8B%E7%82%B9%E7%A7%AF.gif)
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面积/体积与坐标值的关系
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![](../../images/%E6%95%B0%E5%AD%A6/%E3%80%8A%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E3%80%8B%E5%85%8B%E8%8E%B1%E5%A7%86%E6%B3%95%E5%88%99/%E9%9D%A2%E7%A7%AF.gif)
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![](../../images/%E6%95%B0%E5%AD%A6/%E3%80%8A%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E3%80%8B%E5%85%8B%E8%8E%B1%E5%A7%86%E6%B3%95%E5%88%99/%E4%BD%93%E7%A7%AF.gif)
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根据面积关系可以求出Y
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![](../../images/%E6%95%B0%E5%AD%A6/%E3%80%8A%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E3%80%8B%E5%85%8B%E8%8E%B1%E5%A7%86%E6%B3%95%E5%88%99/%E6%B1%82Y.gif)
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X的求取同理
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![](../../images/%E6%95%B0%E5%AD%A6/%E3%80%8A%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E3%80%8B%E5%85%8B%E8%8E%B1%E5%A7%86%E6%B3%95%E5%88%99/%E6%B1%82X.gif)
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上述对XY的求取方式就是克莱姆法则
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在三维下同样适用
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![](../../images/%E6%95%B0%E5%AD%A6/%E3%80%8A%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0%E3%80%8B%E5%85%8B%E8%8E%B1%E5%A7%86%E6%B3%95%E5%88%99/%E4%B8%89%E7%BB%B4.gif)
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