Recommended Math Books

I read lots of math books. Here are some that I highly recommend.

Applied Mathematics

• Stephen Boyd, Lieven Vandenberghe, Applied Linear Algebra

Also known as VMLS (Vectors, Matrices, and Least Squares), as it was known when it was used in draft form at Stanford and UCLA. Let me first explain how this book is atypical It's very applied and not written like a math book (i.e., it doesn't follow the typical definition-theorem-corollary-example pattern). It teaches and uses only the most fundamental of linear algebra over real n-vectors and matrices. There's no mention of the rank-nullity theorem, determinants, dual spaces, or spectral theory.

At this point, pure math guys are probably groaning.

But none of that is the book's intention. The purpose of this book is to teach how to use the basics of linear algebra to solve real world problems. The first two parts of the book provide the rules and "theorems" of basic linear algebra; the third and final part explains how to capitalize on that machinery to perform least squares optimization. Boyd and Vandenberghe provide countless interesting, real-world examples, including among other things linear dynamical systems, signal/image processing, network flows, clustering, portfolio optimization, and classification.

I believe the book's most valuable feature if its excellence at teaching applied math intuition. Anyone who enjoys math knows that intuition is critical. Although I learned almost no new math from this book, I walked away with a great appreciation of how to step away from abstractions to model and solve problems in physical sciences, engineering, finance, and other areas. Since most of my study concentrates pure mathematics, it was a breath of fresh air to see how a little knowledge, a bit of creativity, some elbow grease, and a computer can do some pretty great things!

As an aside, you'll find a very nice third-party supplement here explaining how to implement the tools in this tool with Python.

Algebra

• Serge Lang, Linear Algebra (3rd Ed.)

This is the second of Lang's series on linear algebra, with the first being a much more simple introduction to linear algebra (i.e., one that actually uses numbers and matrices!). Self-studiers like myself will be happy to see that there is an official solutions manual available for this book--a truly priceless resource.

This book is really good for someone who already has some exposure to the basics of linear algebra, such as matrices, simple vector spaces such as R^n and C^n, determinants, and eigenvalues/eigenvectors. Lang's Linear Algebra approaches the topic from a deeper and more abstract level, introducing the reader to other vector spaces and their algebraic structure. It's a very helpful to be exposed to this approach before moving to more advanced topics such as functional analysis, differential geometry, optimization, or quantum mechanics. Lang's theorems are generally concise and easy to follow, and seldom does he unreasonably leave out details or resort to peremptory incantations of "it should be clear." I really enjoyed his treatments of inner product spaces and convex spaces.

There are some sections (such Section 6 of Chapter V on the dual space) that seem hastily written and hard to follow. But in general, it's not difficult to follow the book. The text also has lots of interesting exercises, with the solutions manual easily accessible to check your work. For a self studier who wants a deeper understanding of linear algebra, I don't think you'll find many better resouces.

Topology

• James Munkres, Topology

Munkres' standard book on topology gets a bad rap because it isn't the most engaging of books. I agree with that sentiment. Topology is probably my favorite math subject, and I have trouble seeing how someone can make it dry. Somehow Munkres does precisely that. Notwithstanding its lack of wit and vigor, it is a very comprehensive introduction to point-set and algebraic topology.

Chapter 1 alone is gold. It covers the logical fundamentals of math such as sets, relations, ordering, well-ordered sets, arbitrary products and sequences, mathematical induction, the Axiom of Choice, and Zorn's Lemma. After mastering Chapter 1, you'll have, as Munkres calls it, a "semi-sophisticated" understanding of the basics of mathematics that will serve you well in any other topic.

Chapter 2 is a very complete introduction to the foundations of point-set topology. I really like that Munkres doesn't begin with metric spaces, as many other topology books do. I understand the logic behind focusing on metric spaces first: It provides a gentle path from basic real analysis to the bizarre world of general topological spaces, where intuition can never be trusted. For whatever reason, I liked having to make that leap early on. I had a basic understanding of metric spaces from analysis, and I was amazed to see that topologies could be generated with no concept of a metric or any other concrete way to determine "distance." When Munkres turns to metric spaces toward the end of Chapter 2, I thought to myself, "I see, that's just a specific species of topology. But we don't need no stinkin' metric!"

The rest of Part 1 of the text rounds out the fundamentals of point-set topology. In general, the discussions are easy to follow. Most important to me, the exercises are very interesting and challenging. Since Munkres' book is used in just about every topology class in the world, there are tons of easily-found solutions on the web. That's a godsend to self-studiers!

On a parting note, I love Munkres statements at the beginning of Section 33 on the Urysohn Lemma. He writes, 207 pages into the text:

Now we come to the first deep theorem of the book . . . . [This proof] involves a really original idea, which the previous proofs did not . . . . By and large, one would expect that if one went through this book and deleted all the proofs we have given up to now and then handed the book to a bright student who had not studied topology, that student out to be able to go through the book and work out the proofs independently. (It would take a great deal of time and effort, of course . . . ). But the Urysohn Lemma is on a different level. It would take considerably more originality than most of us possess to prove this lemma unless we were given copious hints!

I can tell you this student would've required copious hints for the first 207 pages of the book, as well!