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 the best of our 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.
Highlights
Cramer’s rule [1] is a formula for solving a system of linear equations as long as the system has a unique solution
Let us choose the partial set of variables as T1, T2, and T3 with j = 3
Using Eq (23), we could express the selected set of variables T1, T2, T3 in terms of the transformed variables, m1a, m2a, m3a, and could keep the remaining set of variables Tk for k ≥ 4 intact. This partial transformation enabled by Cramer’s rule for a partial set of variables in Eq (23) could be used in more general cases involving a large number of variables some of which do not have to be specified in terms of the transformed variables
Summary
Cramer’s rule [1] is a formula for solving a system of linear equations as long as the system has a unique solution. The Dirac delta function δ(x − a) projects out the value of a function f (x) at a certain point x = a after integration:. This elementary property of δ(x) can further be applied to change the variable: y). We derive Cramer’s rule by making use of the Dirac delta function to change the variables of multi-dimensional integrals. A rigorous proof of Cramer’s rule for a partial set of variables by employing mathematical induction is presented in the Appendix
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