The central goals of most introductory linear algebra courses are to develop students' proficiency with matrix techniques, to promote their understanding of key concepts, and to increase their ability to make connections between concepts. In this article, we present an innovative method using adjacency matrices to analyze students' interpretation of and connections between concepts. Three cases provide examples that illustrate the usefulness of this approach for comparing differences in the structure of the connections, as exhibited in what we refer to as dense, sparse, and hub adjacency matrices. We also make use of mathematical constructs from digraph theory, such as walks and being strongly connected, to indicate possible chains of connections and flexibility in making connections within and between concepts. We posit that this method is useful for characterizing student connections in other content areas and grade levels.
Natalie E. Selinski, Chris Rasmussen, Megan Wawro and Michelle Zandieh
In a recent issue of the Mathematics Teacher, Nicolai (1974) detailed several “discoveries” in linear algebra that were made by him and one of his classes. A look at their ideas yielded several discoveries showing that the relationships described by Nicolai are special cases of more general, interesting situations.
Michael B. Nicolai
An eighth-grade teacher and his class share the excitement of discovery.
Thomas J. Brieske
The unifying power of vector space concepts can be developed with rich geometric intuition and practical payoff following this guided tour through linear equations.
Murli M. Gupta
A general solution of a problem in linear algebra.
Aina Appova and Tetyana Berezovski
Students connect operations in linear algebra with geometric representations.
Rex H. Shudde
Linear programming is one of the tools of operations research which has been developed in the past two decades. It is a natural extension of linear algebra.
In introductory linear algebra courses one continually seeks interesting sets of matrices that are closed under the operations of matrix addition, scalar multiplication, and if possible, matrix multiplication. Most texts mention symmetric and antisymmetric matrices and ask the reader to show that these sets are closed under matrix addition and scalar multiplication but fail to be closed under matrix multiplication. Few textbooks, if any, suggest an investigation of the set of matrices that are symmetric with respect to both diagonals, namely bisymmetric matrices. The following is a sequence of relatively straightforward problems that can be used as homework, class discussion, or even examination material in elementary linear algebra classes.
Samuel L. Dunn and Lawrence W. Wright
The field of economics provides many opportunities for applying mathematics. The quantification of economics that has occurred in the last thirty years has made it necessary that economists be trained in the uses of higher mathematics. Algebra, geometry, calculus, probability theory and statistics, higher analysis, linear algebra, and computer science are some of the tools being used in contemporary approaches to economics.
Arthur T. Benjamin and Jennifer J. Quinn
We fully concur with Richard Askey's February 2004 “Delving Deeper” column. Discovering and proving identities containing Fibonacci numbers can be satisfying for students and teachers alike. His article touched on multiple strategies including induction, linear algebra, and a hefty dose of algebraic manipulation to derive many interesting identities. However, a single method can be employed to explain all of these identities more concretely, leading to deeper understanding and intuition. We are referring to the method of combinatorial proof.