Definition of echelon form of a linear system

Theorem: All echelon forms of a given linear system have the same pivot positions, and therefore the same free variables.

While the exact numerical entries in different echelon forms may vary (depending on the sequence of row operations), the positions of the pivots are uniquely determined by the system itself. This means the classification of variables into basic and free variables is invariant.

Why this matters: When solving a system, you can use any valid sequence of row operations — the set of free variables you identify will always be the same.

Note: The reduced row echelon form (RREF) is unique for a given matrix, but ordinary echelon forms are not unique.