Using the sawtooth map as the basis of a coupled map lattice enables simple analytic results to be obtained for the global Lyapunov spectra of a number of standard lattice networks. The results presented can be used to enrich a course on chaos or dynamical systems by providing tractable examples of higher dimensional maps and links to a number of…

This paper discusses a project for linear algebra instructors interested in a concrete, geometric application of matrix diagonalization. The project provides a theorem concerning a nested sequence of tetrahedrons and scaffolded questions for students to work through a proof. Along the way students learn content from three-dimensional geometry and…

Linear algebra is best done with block matrices. As evidence in support of this thesis, we present numerous examples suitable for classroom presentation.

Almost all finite groups encountered by undergraduates can be represented as multiplicative groups of concise block-diagonal binary matrices. Such representations provide simple examples for beginning a group theory course. More importantly, these representations provide concrete models for "abstract" concepts. We describe Maple lab…

We present an algorithm for a matrix-based Enigma-type encoder based on a variation of the Hill Cipher as an application of 2 × 2 matrices. In particular, students will use vector addition and 2 × 2 matrix multiplication by column vectors to simulate a matrix version of the German Enigma Encoding Machine as a basic example of cryptography. The…

This linear algebra note offers teaching and learning ideas in the treatment of the inner product space R[superscript m x n] in a technology-supported learning environment. Classroom activities proposed in this note demonstrate creative ways of integrating MATLAB technology into various properties of Frobenius inner product as visualization tools…

A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by…

We present a short-term class project used in an introductory linear algebra course, designed to engage students in matrix algebra. In this activity, students responded to a survey of their pop culture tastes. Using the survey responses, they worked to design a series of matching algorithms, using matrices, with the goal of matching the students…

A stochastic matrix is a square matrix with nonnegative entries and row sums 1. The simplest example is a permutation matrix, whose rows permute the rows of an identity matrix. A permutation matrix and its inverse are both stochastic. We prove the converse, that is, if a matrix and its inverse are both stochastic, then it is a permutation matrix.

An elementary proof using matrix theory is given for the following criterion: if "F"/"K" and "L"/"K" are field extensions, with "F" and "L" both contained in a common extension field, then "F" and "L" are linearly disjoint over "K" if (and only if) some…

A certain weighted average of the rows (and columns) of a nonnegative matrix yields a surprisingly simple, heuristical approximation to its singular vectors. There are correspondingly good approximations to the singular values. Such rules of thumb provide an intuitive interpretation of the singular vectors that helps explain why the SVD is so…

This paper deals with the ancient origin of matrices, and the system of linear equations. Included are algebraic properties of matrices, determinants, linear transformations, and Cramer's Rule for solving the system of algebraic equations. Special attention is given to some special matrices, including matrices in graph theory and electrical…

This paper introduces a new matrix tool for the sowing game Tchoukaillon, which establishes a relationship between board vectors and move vectors that does not depend on actually playing the game. This allows for simpler proofs than currently appear in the literature for two key theorems, as well as a new method for constructing move vectors.We…

This article outlines a problem-centered approach to the topic of canonical matrix forms in a second linear algebra course. In this approach, abstract theory, including such topics as eigenvalues, generalized eigenspaces, invariant subspaces, independent subspaces, nilpotency, and cyclic spaces, is developed in response to the patterns discovered…

Martin Gardner wrote about a coin-flipping trick, performed by a blindfolded magician. The paper analyses this trick, and compares it with a similar trick using three cups flipped in pairs. Several different methods of analysis are discussed, including a graphical analysis of the state space and a representation in terms of a matrix. These methods…