Linear inhomogeneous system/Elimination/Echelon form/Fact/Proof
Proof
This follows directly from the elimination lemma, by eliminating successively variables. Elimination is applied firstly to the first variable (in the given ordering), say , which occurs in at least one equation with a coefficient (if it only occurs in one equation, then this elimination step is already done). This elimination process is applied as long as the new subsystem (without the working equation used in the elimination step before) has at least one equation with a coefficient for one variable different from . After this, we have in the end only equations without variables, and they are either only zero equations, or there is no solution.
