Linear Algebra by Jim Hefferon

What’s Linear Algebra About?

When I started teaching linear algebra I found three kinds of texts. There were applied mathematics books that avoid proofs and cover the linear algebra only as needed for their applications. There were advanced books that assume that students can understand their elegant proofs and understand how to answer the homework questions having seen only one or two examples. And there were books that spend a good part of the semester doing elementary things such as multiplying matrices and computing determinants and then suddenly change level to working with definitions and proofs.

Each of these three types was a problem in my classroom. The applications were interesting, but I wanted to focus on the linear algebra. The advanced books were beautiful, but my students were not ready for them. And the level-switching books resulted in a lot of grief. My students immediately thought that these were like the calculus books that they had seen before, where there is material labelled `proof’ that they have successfully skipped in favor of the computations. Then, by the time that the level switched, no amount of prompting on my part could convince them otherwise and the semesters ended unhappily.

That is, while I wish I could say that my students now perform at the level of the advanced books, I cannot. However, we can instead work steadily to bring them up to it, over the course of our program. This means stepping back from rote computations of the application books in favor of an understanding of the mathematics. It means proving things and having them understand, e.g., that matrix multiplication is the application of a linear function. But it means also avoiding an approach that is too advanced for the students; the presentation must emphasize motivation, must have many exercises, and must include problem sets with many of the medium-difficult questions that are a challenge to a learner without being overwhelming. And, it means communicating to our students that this is what we are doing, right from the start.

Summary Points

The coverage is standard: linear systems and Gauss’ method, vector

  spaces, linear maps and matrices, determinants, and eigenvectors and
  eigenvalues. The Table of Contents gives you a quick overview.

Prerequisites A semester of calculus. Students with three semesters

  of calculus can skip a few sections.

Applications Each chapter has three or four discussions of

  additional topics and applications. These are suitable for
  independent study or for small group work.

What makes this book different? Its approach is

  developmental. Although the presentation is focused on proving
  things and covering linear algebra, it does not start with an
  assumption that students are already able at abstract work. Instead,
  it proceeds with a great deal of motivation, and many examples and
  exercises that range from routine verifications to (a few)
  challenges. The goal is, in the context of developing the usual
  material of an undergraduate linear algebra course, to help raise
  the level of mathematical maturity of the class.