Elementary row operations, Gaussian elimination, and Gauss-Jordan reduction play key roles in an introductory linear algebra course. While some form of geometric visualization typically accompanies the introduction of these procedures, textbooks tend to focus nearly exclusively on their algebraic aspects. In this article, I propose a dynamic geometric illustration of elementary row operations, which can help students better understand Gaussian elimination and Gauss-Jordan reduction, from both mechanical and conceptual points of view.
Przemyslaw Bogacki is an Associate Professor of Mathematics at Old Dominion University .
In an introductory linear algebra course, elementary row operations are a recurring theme. They are typically introduced in the context of Gauss-Jordan reduction and/or Gaussian elimination and are repeatedly referred to when discussing a variety of topics, including matrix inverses, determinants, linear independence, coordinates, eigenvectors, and linear transformations (Carlson et al., 1993).
Almost all linear algebra textbooks incorporate at least some discussion of the geometry of vectors and lines in R2, as well as vectors, lines, and planes in R3 . Specifically, there are renderings of planes in R3 corresponding to a system of equations in three unknowns (in various configurations: one solution, two parallel planes, etc.). See, e.g., Kolman and Hill (2001, p. 6) and Bretscher (2001, pp. 3-4). However, the mode in which elementary row operations are discussed in textbooks is almost exclusively algebraic.
Some authors have written modules, usually created on CAS platforms such as Maple or Mathematica, designed to overcome this limitation by illustrating the geometry of each iteration of Gaussian elimination (e.g., Symancyk, undated). I will take another step in this direction. Rather than presenting the iterations in a "discrete" manner, I will illustrate dynamically the transition corresponding to each row operation in a "continuous" fashion, using an animation-based approach. I believe this approach can help students better understand both the mechanics of how these operations are executed (e.g., "how we choose the pivot") and the motivation for the entire procedure of Gauss-Jordan reduction or Gaussian elimination (i.e., why we try to get these planes aligned with the coordinate axes).
To accomplish these goals, I have designed an activity called "HINGES" -- the acronym will be explained on page 3.
Published May, 2005
© 2005 by Przemyslaw Bogacki