Proof of Cramer's rule with Dirac Delta Function

June-Haak Ee; Jungil Lee; Chaehyun Yu

doi:10.1088/1361-6404/aba455

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arXiv · 2006.01609

Proof of Cramer's rule with Dirac Delta Function

June-Haak Ee·Jungil Lee·Chaehyun Yu

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Abstract

We present a new proof of Cramer's rule by interpreting a system of linear equations as a 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. As a byproduct, we derive a generalized version of Cramer's rule that applies to a partial set of variables, which is new to our best knowledge. 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.

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June-Haak Ee, Jungil Lee, Chaehyun Yu. 2020-05-25. Proof of Cramer's rule with Dirac Delta Function. https://doi.org/10.1088/1361-6404/aba455

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