Linear Methods: A General Education Course

The MAA’s 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences includes the Content Recommendation “Mathematical sciences major programs should include concepts and methods from calculus and linear algebra.” The report points out that linear algebra “introduces students to discrete mathematics, algorithmic thinking, a modicum of abstraction, moderate sophistication in notation, and simple proofs.” In typical undergraduate curricula, linear methods arise as needed in pre-calculus algebra and analysis courses to be first treated with the full attention of a semester-long focus as part of the first year in calculus. The fundamentals text here is a course with little in the way of mathematical prerequisites “aside from some familiarity with simple concepts from high school algebra and geometry.” Here is an opportunity to present elementary linear algebra in some depth without requiring the college algebra fundamentals often presented a year or more beforehand. The authors seek to provide this thorough foundation in linear methods also to non-math majors and have a considered aim to also instruct in “the nature of mathematical proof.” A companion site has supplementary material in this area and an instructor’s solution manual is available.

If not used as a standalone course, this text still offers opportunities to amplify linear methods in a college algebra course. This includes the extensive coverage of Gauss-Jordan around elementary row operations with little reliance on matrix inverses. The introduction to linear regression is based on transpose matrices over the iterative least squares sum approach typically used at this level. I find Algebra I textbooks often leave the transpose operation out there as a seldom applied technique while students find least squares a tedious exercise better done by a calculator and offering no real insight. I find intriguing the idea of introducing linear regression as a matrix method applying newly introduced matrix operations.

This textbook introduces elementary graph theory, Markov chains, least-squares polynomials, geometric transformations, orthogonality, and more. Here a student taking a course or studying independently the self-contained text has exercises, solutions to odd-numbered problems, and chapter “glossaries” which are recaps of key concepts and terms acting as a chapter summary.

Tom Schulte is the mathematics instructor for Upper Iowa University's New Orleans Center.