Linear Algebra

by
Jim Hefferon
Mathematics
Saint Michael's College
Colchester, Vermont USA 05439

This text Linear Algebra is free for downloading. It covers an undergraduate first course, with a prerequisite of Calculus I. You can use it either as a main text, or as a supplement to another text, or for independent study. A separate file with all answers, fully worked, is also freely downloadable.

What's Linear Algebra about?

When I started teaching this subject I found three kinds of texts. There were applications 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 know 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 multiplying matrices and computing determinants and then suddenly change level to working with definitions and proofs.

In my classroom each of these types was a problem. 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: students estimated that these were like calculus books, where there is material labelled `proof' that can skipped in favor of computations, and when the level switched no amount of prompting by me could convince them otherwise.

That is, my students cannot now perform at the level assumed by the advanced books. But my goal is to work steadily to have them come up to that level over the undergraduate program. This course is a great place to make progress on this goal.

This goal leads straight to a number of tasks. It means first that we must prove things. It means also that we must step away from the rote computations of the applications books in favor of understanding the concepts (for instance, students must understand matrix-vector multiplication as representing the application of a linear function). But, it means also being sure that the approach is not too advanced for the current level of the students: the presentation must emphasize motivation and naturalness, have many examples, and have many exercises, particularly the medium-difficult questions that challenge a learner without overwhelming them.

Summary Points

These are the chief features of Linear Algebra.

• 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.
• Prerequisite: one semester of calculus. Students with three semesters of calculus can skip a few sections.
• Applications. Each chapter ends with 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.
• License. This text is Free. Use it under either the GNU Free Documentation License or the Creative Commons Attribution-ShareAlike 2.5 License, at your discretion.

  Note to bookstores: in particular, instructors have permission to make copies of this material, either electronic or paper, and sell those copies to students. Many schools use this text in this way. If you have further questions, please feel free to contact me.

Here is Linear Algebra

• The whole book Linear Algebra along with the answers to all exercises. If you save these two files in the same directory then clicking on an exercise will send you to its answer and clicking on an answer will send you to its exercise. This does not work on the web, it only works if you download the files.
• The source. You need to know LaTeX and MetaPost to work with it, although there is a readme file to get you going. Also there is a bit of optional material.

I made it Free because I'm delighted to have it on student's desks, doing what I wrote it for. Around here, running off double-sided copies and comb-binding them costs about $20.00. I'd be glad for any comments, especially reactions from instructors who have tried it out.

Some notes

• The files are current as of 2006-June-20.
• I sometimes get emails from people who have trouble downloading. My logs show hundreds of successful downloads every day so it is not me, it is you. I'm sorry; I cannot email you the files. The only thing I can suggest is that if you have Windows you can try to add me to your list of trusted sites. If that doesn't work then you simply need to ask your support person. If it does work, and you find it astonishing that you are using a system that takes me as not to be trusted, then you might want to try another way.
• There is only one review that I know of.
• If you make these materials available elsewhere, please also make available a link to this page, so that your users can get the most recent version.
• Do not send me questions about the subject. I simply delete those mails.

Can You Help With Linear Algebra?

In addition to the book, the answers to the exercises, and the source, you can get some materials that were contributed by others. If you in turn have something that you are able to share please write me. Anything is welcome, from a suggestion about a better way to prove a result or make an example, to an entire Topic or subsection. In particular, your used exams or problem sets would be very welcome, especially if you can contribute the TeX or LaTeX source, so that a user could cut or paste. Several people have asked me about a Topic on eigenvectors and eigenvalues in Quantum Mechanics. Sadly, I don't know any QM. If you can help, that'd be great.

I receive typo and bug reports from many people and they are a big help. I save them and do updates in a batch so if you made a report and it hasn't gotten addressed yet, please be patient.

Naturally, all contributions will be gratefully acknowledged (or anonymous, if you like).

Has Linear Algebra Helped You?

If you find the book useful, drop me a note in the mail. In particular, if you use it in a class, a letter on letterhead to the address below would be great.

   Jim Hefferon
   Mathematics, Saint Michael's College
   One Winooski Park
   Colchester, VT USA, 05439 

Page Information

The page above. The page maintainer is Jim Hefferon.

This site Joshua is located in the Mathematics Department of Saint Michael's College in Colchester, Vermont USA. Joshua runs under Linux. Open Source software is a great idea. This project would not have gotten done without it.

(Credit for the logo to Matt Ericson.)