During my undergraduate education, I came across a number of different textbooks, none of which really satisfied me. That is until one of my undergraduate professors recommended Linear Algebra by Hoffman and Kunze, which I finally found time to dig into during Dr. Healey’s linear algebra lectures at the Institute for Advanced Analytics (IAA). I am grateful to my professor for having recommended Linear Algebra, and I think other students should use it as a resource to refresh their memory.
There are two chief virtues of Hoffman and Kunze’s book. First, it is succinct in its exposition. The authors write all of the things you need to read and few of the things you don’t. Second, the book does not lack for detail. Theorems are proved, and the book is not afraid to detail the important corollaries of the main results. The result is a reasonably comprehensive treatment of foundational linear algebra that does not bore the reader nor demand too much of them.
Of course, Hoffman and Kunze aren’t the only authors to write a Linear Algebra textbook. One text of considerable renown is Axler’s Linear Algebra Done Right. The cardinal sin of Linear Algebra Done Right in the eyes of some is its oft-discussed postponement of the determinant until the very end. My biggest gripe with the most recent edition is its visual presentation. There is no reason for the page that defines an eigenvalue to include four different colors, a fun fact about the term itself, and a remark inside of a proposition instead of in its own environment. By the way, the proposition should be labeled as such. For a student who grew up on Axler’s book, these might be charming quirks. For me, it renders the book difficult to read.
Figure 1: Page from Linear Algebra Done Right
Treil’s Linear Algebra Done Wrong is something of a counterpoint to Axler’s book. It has a more understated visual style more in line with a typical math textbook. The text also follows a more typical development in which determinants play a more central role. Ultimately, I find that Linear Algebra Done Wrong’s text is too verbose for use by someone with prior knowledge. However, the text has a lot of merit for a student who is new to the subject.
Organizationally, Treil’s final chapter on “advanced” spectral theory suffers from being so far from the chapter on ordinary spectral theory. On the other hand, Hoffman and Kunze flow quite directly from their discussion of eigenvalues to invariant sub-spaces and on into a chapter on canonical forms. I felt that this organization made the main ideas easier to follow. Moreover, I find the construction of the determinant they give much easier to follow than the construction presented in Linear Algebra Done Wrong.
To its credit, Linear Algebra devotes three sections at the end of chapter 3 to the dual of a vector space. This comes well before any discussion of inner product spaces. I found that this organization, paired with the lucid writing, helped me to situate my understanding of concepts from inner product spaces in a broader context.
Another point in the book’s favor is its willingness to introduce the terminology of modern algebra. This simplifies the exposition for the reader who has seen abstract algebra while furnishing the reader who has not with a brief introduction. The invocation of abstract algebra helps in the study of vector spaces other than Cn or Rn. Of note, the vector space of linear transformations on a vector space is mentioned as more than a cursory example, as it should be given the usefulness of this perspective when thinking about singular value decomposition.
Another benefit of grappling with abstract algebra is the book’s ability to give a meaningful discussion of the algebra of polynomials before it is required for spectral theory in chapter 6. This streamlines the exposition of more advanced spectral theory considerably. Not only this but the authors explain the parts of spectral theory beyond mere diagonalization quite elegantly.
Linear Algebra’s greatest vice is delaying inner product spaces until chapter 8. The result is that the spectral theorem is not proven until almost the end of the book. Yet for a student who is reviewing the subject, this is no great difficulty. Moreover, to move it earlier would inevitably displace currently well-placed content.
One vice particularly relevant to a data science audience is that there is no discussion of singular value decomposition. However, a reasonably comprehensive discussion of singular value decomposition’s usefulness requires studying the metric structure of vector spaces of linear transformations/matrices. Taking this project seriously would be another textbook in itself. In fact, there is just such a book. The interested reader can see Horn and Johnson’s Matrix Analysis. A more elementary exposition appropriate for a wider audience can be found in Treil’s book, and this is the sort of exposition that I would have liked to see in Linear Algebra.
Although it’s not without flaws, I found Hoffman and Kunze’s overall approach edifying. It may not be appropriate for those who have never taken a rigorous mathematics course. However, if you have experience with mathematics and want to relearn linear algebra (perhaps because you enroll in the IAA’s MSA program), I highly recommend opening up Hoffman and Kunze’s Linear Algebra.
Columnist: Evan Hunter