Iterative decreasing dimension algorithm (Q2458721)

A system of \(n\) linear equations \(Ax=f\) is split into the first equation and a system of the remaining \(n-1\) equations. Solve the first equation for the first component \(x_k\) of \(x\) that has a nonzero coefficient \(a_{1k}\). This \(x_k\) will depend upon the other \(n-1\) components of \(x\), but the first equation is hereby satisfied. Then plug this solution into the second part of the system which has now become a system of \(n-1\) equations with \(n-1\) unknowns, and the procedure can be repeated. Hence, the dimension of the system decreases in every (elimination) step. This is basically another way of writing Gaussian elimination.