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+---
+title: "《线性代数》克莱姆法则"
+date: 2023-08-11T21:07:47+08:00
+
+---
+
+## 克莱姆法则
+
+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
+
+
+对于一个线性方程组
+$$
+\begin{cases}
+    3x+2y &=-4 \\\\ 
+    -1x+2y &=-2 
+\end{cases}
+$$
+
+可以将其看作对向量的一个已知的矩阵变换,且结果已知
+
+$$
+\begin{bmatrix}
+    3 & 2 \\\\
+    -1 & 2
+\end{bmatrix}
+\begin{bmatrix}
+    x \\\\
+    y
+\end{bmatrix} = 
+\begin{bmatrix}
+    -4 \\\\
+    -2
+\end{bmatrix}
+$$
+
+![](../../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)
+
+$$
+x\begin{bmatrix}
+    3  \\\\
+    -1 
+\end{bmatrix} +
+y\begin{bmatrix}
+    2 \\\\
+    2
+\end{bmatrix} = 
+\begin{bmatrix}
+    -4 \\\\
+    -2
+\end{bmatrix}
+$$
+
+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
+
+
+![](../../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)
+
+面积/体积与坐标值的关系
+
+![](../../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)
+
+![](../../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)
+
+根据面积关系可以求出Y
+
+![](../../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)
+
+X的求取同理
+
+![](../../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)
+
+上述对XY的求取方式就是克莱姆法则
+
+在三维下同样适用
+
+![](../../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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