We present a new proof of Cramer's rule by interpreting a system of linear equations as transformation of $n$-dimensional Cartesian-coordinate vectors. To find the solution, we carry out the inverse transformation by convolving the original coordinate vector with Dirac delta functions and changing integration variables from the original coordinates to new coordinates. Our formulation of finding a transformation rule for multi-variable functions shall be particularly useful in changing a partial set of generalized coordinates of a mechanical system.

中文翻译：

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用狄拉克 delta 函数证明克莱默规则

我们通过将线性方程组解释为 $n$ 维笛卡尔坐标向量的变换，提出了 Cramer 规则的新证明。为了找到解决方案，我们通过将原始坐标向量与 Dirac delta 函数进行卷积并将积分变量从原始坐标更改为新坐标来执行逆变换。我们为多变量函数寻找转换规则的公式在改变机械系统的部分广义坐标集时特别有用。