My college linear algebra course was held early in the morning, and it was devoted almost entirely to blackboard proofs. The professor would stand in front of the room, half asleep, and write:

“Theorem. Lemma. Lemma. Proof. Theorem…”

Despite this experience, I somehow managed to learn about eigenvectors and kernels. Or at least, I learned how to write proofs about them. But I had no intuition for linear algebra: I couldn’t visualize it, and I couldn’t explain why anybody, anywhere, ever cared about eigenvectors.

Years later, in a computer vision class, I finally learned to care about linear algebra. It could solve all sorts of cool problems! (Eigenfaces, in particular, blew me away.) And since then, I’ve encountered linear algebra everywhere. But my intuition is still piecemeal, built from half-a-dozen applications over the years.

My motto for math is, “If it keeps showing up, build a rock-solid intuition for how it works.” And towards that end, I’ve been looking for a good linear algebra textbook.

My ideal linear algebra textbook would:

1. Include plenty of motivating examples.
2. Show how to solve real-world problems.
3. Devote plenty of time to proofs.

The proofs, after all, are necessary in the real world. If you ever attempt to do something slightly odd, you’ll want to prove that it actually works.

### Jim Hefferon’s Linear Algebra

Professor Jim Hefferon’s Linear Algebra is available as a free PDF download. But don’t be fooled by the price: Hefferon’s book is better than most of the expensive tomes sold in college bookstores.

Everything in Hefferon’s book is superbly motivated. The first chapter begins with two real-world examples: Unknown weights placed on balances, and the ratios of complex molecules in chemical reactions. These examples are used to introduce Gauss’s method for solving systems of linear equations. Further into the book, the examples begin to tie back to earlier chapters. Determinants, for example, are motivated by the usefulness of recognizing isomorphisms and invertible matrices.

But Hefferon’s emphasis on real-world examples is admirably balanced by an abundance of proofs. The first proof appears on page 4, and nearly everything is proven either in the main text or in the exercises. This will be helpful for readers who (like me) are trying to bring more rigor to their mathematical thinking.

### The “Topics”: Fascinating real-world problems

The most delightful part of the book, however, are the “Topics” at the end of each chapter. These cover a wide range of fields, including biology, economics, probability and abstract algebra. The topic “Stable Populations” begins:

Imagine a reserve park with animals from a species that we are trying to protect. The park doesn’t have a fence and so animals cross the boundary, both from the inside out and in the other direction. Every year, 10% of the animals from inside of the park leave, and 1% of the animals from the outside find their way in. We can ask if we can find a stable level of population for this park: is there a population that, once established, will stay constant over time, with the number of animals leaving equal to the number of animals entering?

Hefferon relates the solution to Markov chains and eigenvalues, cementing several important intuitions firmly in place.

Other topics include basic electronics, space shuttle O-rings, and the number of games required to win the World Series. There are plenty of CS-related discussions, too: a survey of things that can go wrong in naive numeric code, the time required to calculate determinants, and how the memory hierarchy affects array layout.

Hefferon’s love for linear algebra is infectious, and his “Topics” will appeal to anybody who does recreational math.

### “Free” as in “freedom”

Linear Algebra is published under the GNU Free Documentation License and the Creative Commons Share Alike license.

What this means: You may make copies of the book, or even print them out at a copyshop and charge students a fee. You may also create a custom version of the textbook, and share it with anybody who’s interested. The only restriction: You must “share alike,” honoring the original author’s terms as you pass along the textbook.

### Miscellaneous notes

Hefferon has put out a call for extra material. In particular, he’d love to have a section on quantum mechanics:

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.

On the downside, the internal PDF links in Linear Algebra are broken in MacOS X Preview. This is odd, because the LaTeX hyperref package usually works fine with Preview.

The reddit discussion of Linear Algebra has pointers to several other linear algebra textbooks, with varying emphasis. And many other free math textbooks are available online.

If you have any favorite math books (paper or PDF, for any area of mathematics), please feel free to recommend them in the comment thread!