A Column-Vector Geometric Interpretation of Linear Systems and Gaussian Elimination

Humberto José Bortolossi; Luciana Prado Mouta Pena

Archives · Vol. 14, No. 1 (2026)

A Column-Vector Geometric Interpretation of Linear Systems and Gaussian Elimination

Humberto José Bortolossi, Luciana Prado Mouta Pena

Humberto José Bortolossi — Institute of Mathematics and Statistics, Fluminense Federal University

Luciana Prado Mouta Pena — Department of Mathematics, Federal University of Juiz de Fora

↓ Download PDF pp. 27–36

Abstract

This paper presents GeoGebra constructions that provide a visual and interactive interpretation of linear systems and Gaussian elimination from the vector-and-matrix viewpoint of linear algebra. The approach emphasizes the column-vector interpretation of Ax=b: solving a linear system means expressing the right-hand side as a linear combination of the columns of the coefficient matrix. In this setting, a fundamental row operation in Gaussian elimination has a natural geometric counterpart: a shear transformation. This perspective makes visible the geometric meaning of row reduction and clarifies its connection with the LU factorization of matrices. The GeoGebra applets support dynamic exploration of the links among algebraic procedures, vector combinations, matrix operations, and geometric transformations.

Keywords: linear systems, GeoGebra, shear transformations, Gaussian elimination, LU factorization

GeoGebra Materials

Licensed under CC BY-NC-SA 4.0 · ISSN 2162-3856 · North American GeoGebra Journal