Linear systems: a column‐space approach to Cramer's rule

Abstract

This paper deals with a simple method for the direct solution of linear systems which enables a geometric approach to Cramer's rule. In order to get the explicit representation of a certain unknown, the n‐dimensional column space of the system matrix will be decomposed into a (n – l)‐dimensional subspace and its orthogonal complement. Each unknown has exactly one corresponding decomposition. Every considered unknown can be computed as the quotient of two inner products where the numerator is formed by an arbitrary vector, say W, of the respective orthogonal complement together with the right‐hand side and the denominator by W and the column vector corresponding to the unknown. This leads to a special version of the inverse of the system matrix. It will be shown that the indicated representation of the unknowns is equivalent to Cramer's rule. Moreover, by means of this equivalence a relationship to the QR factorization via Gram‐Schmidt orthonormalization will be pointed out.