A Gauche Perspective on Row Reduced Echelon Form and Its Uniqueness

Using a left-to-right "sweeping" algorithm, we define the Gauche basis for the column space of a matrix M. Interpreting the row reduced echelon form (RREF) of M by Gauche means gives a direct proof of its uniqueness. A corollary shows that the (right) null space of M determines its row equivalence class, unmasks a sanitized version of the assertion "if two systems are solution equivalent they are row equivalent," and presents the null space as a distinguished graph. We conclude with pedagogical reflections.