No Bullshit Guide to Linear Algebra, Savov (2017)

It may be best to begin with this book’s weakest element, and that is the title: No Bullshit Guide to Linear Algebra. The title implies that this book is fluffy, that it is not a serious book on mathematics; and that is not the case. I hesitated to buy this book because of its title.

This is a book on linear algebra, the branch of mathematics whose central object is the matrix. Of its nine chapters, the first chapter—the longest—covers algebra of the sort the reader likely studied in high school. If your knowledge of the fundamentals is sound, you can safely skip over this chapter. But don’t do that. Why not? There are several reasons. First, you may find something in there that you don’t already know. You don’t know everything. You wouldn’t want to miss it. Also, it can be useful to see a new perspective, to observe from a new vantage point how things all fit together. The author is disciplined and thoughtful in his use of notation and the first chapter is where things are first introduced. But, aside from all that, the text is genuinely well written. Good writing is a pleasure to read and there is no reason to deny yourself.

The next five chapters present the contents of linear algebra proper. This treatment is thin on rigorous proof. But, rigorous proofs are not particularly useful for creating comprehension and building intuition. What you do find here are well-crafted descriptions and thoughtful explanations, many of a very high quality. Some give a uniquely useful perspective. Also, despite its only taking 230 pages, this material goes beyond the basics and includes advanced topics, i.e., fancy stuff.

The last three chapters cover applications with one chapter devoted to probability and the final chapter to quantum mechanics. Fourier transforms are covered in the catch-all chapter. If a book’s ending is its destination, then that destination would be quantum computing and quantum information theory. That may not be the reader’s destination and that’s fine. As a personal note, I had read Feynman’s original quantum computing paper some time ago, but it didn’t really come together for me until Savov supplied the missing pieces that Feynman had not made clear—or at least not clear enough to me.

I must admit that, in these final chapters, the author sometimes allows himself a fig leaf’s worth of coverage for a volume’s worth of subject. Thus, it becomes harder to read in places and I didn’t always come away with the sense of crystal clarity that I took away from the earlier chapters. The author gives many pointers out for further exploration. In fact, those appear throughout the book.

If there were to ever be a third edition, any revision should certainly proceed from back to front. There are elements early in the book that should be trapped in amber and preserved for eternity, just as they are. But, as for the last section? If the applications section were expanded, perhaps tripled, so as to provide for longer explanations, I’m not sure it would still be a linear algebra book. The applications section already runs long, relative to the central part of the book. Likewise, removing any of this in the interest of brevity would be a tragedy. So, it is as it is meant to be.

I hope there are more works forthcoming from Ivan Savov.