A Gauche Perspective on Row Reduced Echelon Form and Its Uniqueness

Abstract

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.

Acknowledgment

The Gauche idea emerged from a conversation with Professor Gilbert Strang in the fall of 2019 at MIT’s Endicott House. The author is grateful to Professor Strang for the conversation and for his inspiring writings through the years. He is also grateful to MIT for the invitation to the Endicott House event, and he would happily repeat the experience. In addition, he thanks all who have commented on the paper, especially Mehmet Orhon, Géza Schay, and Dennis Wortman.

Notes

1 Fifth column—a group of secret sympathizers or supporters of an enemy that engage in espionage or sabotage within defense lines or national borders—Merriam-Webster dictionary

Additional information

Notes on contributors

Eric L. Grinberg

Eric L. Grinberg began his study of linear algebra in the summer before his freshman year at Cornell, when Oscar S. Rothaus challenged him to read Halmos’s Finite Dimensional Vector Spaces and face an oral exam. A reading course with R. Keith Dennis followed, using Hoffman and Kunze’s Linear Algebra. ELG went on to write a thesis on Radon transforms in compact symmetric spaces, under the direction of Victor Guillemin and Shlomo Sternberg, from whom he continues to draw inspiration. His research interests are in analysis and geometry, especially in the context of group symmetry, with a focus on integral geometry. He taught at the University of Michigan, Temple University, Brooklyn Poly, the University of New Hampshire and, since 2010, at the University of Massachusetts Boston. He has served as associate dean, and as department chair with multiplicity.