The Oxford Linear Algebra for Scientists

Typically, when I see a phrase like “for scientists” in the title of a book, I immediately conclude that the book puts mathematical rigor on the back burner in favor of stressing applications from a computational “how to” point of view. It turns out, however, that that is not the case with the book now under review, which offers an introduction to linear algebra that is characterized by precise definitions and rigorous proofs as well as an indication of how linear algebra actually finds use in other areas of mathematics and science.

 

In fact, the book is somewhat more abstract than most undergraduate linear algebra texts: it works with arbitrary fields rather than just the real and complex numbers. Thus, it begins with chapters on groups and fields before vector spaces are even defined. On the other hand, the focus is on finite-dimensional spaces, so fancy set theoretic tools like Zorn’s Lemma are not invoked.

 

After a brief discussion of linearity that sets the stage for what is to follow, the book breaks up into eight parts. The first part is preliminary and has a chapter on sets and functions, as well as the aforementioned chapters on groups and fields. Part 2 discusses vector spaces by first introducing coordinate spaces. Bases and dimension are covered here. The next part of the book discusses the dot product, cross product and scalar triple product, with applications to coordinate geometry. Linear mappings are the subject of part 4, along with their representation by matrices. Now that matrices have been introduced, they are used in part 5 on systems of linear equations. Determinants are also the subject of a chapter in this part. Part 6 discusses eigenvalues and eigenvectors, starting from the definition but proceeding up diagonalization, the characteristic and minimal polynomials, the Cayley-Hamilton theorem, and the Jordan form. The next part of the book discusses inner product spaces on real and complex spaces and the linear operators defined on these spaces (Hermitian, normal, etc.). Bilinear and sesquilinear forms are also discussed. Finally, part 8 of the book contains two chapters, one on the dual space of a vector space and the other on tensors.

 

Interspersed throughout the book are two dozen vignettes, each about a page or two long and discussing applications of linear algebra, both to other branches of mathematics (such as graph theory, cryptography and differential equations) and to various other fields of science (for example, neural networks, quantum computing, and data compression). Though not as detailed and rigorous as the rest of the book, these vignettes do give some indication of how linear algebra shows up elsewhere.  A helpful chart of all these applications is included in the text.

 

I was puzzled, however, by the omission of some topics from the text. In view of the fact that projection matrices are discussed, the inclusion of a section on least squares approximation would have seemed a natural thing to include. Other topics that are missing that one might perhaps have expected to find in a textbook “for scientists” are eigenvalue calculation, matrix norms and the condition number, Markov matrices, and positive matrices. 

 

These omissions notwithstanding, however, this is an interesting book. It is well-written, with many examples and worked out problems. Every chapter ends with a reasonable assortment of exercises, most of which struck me as being on the easy end of the spectrum. It starts from scratch but covers some topics in linear algebra that are typically thought of as advanced. Anyone who teaches, or is interested in, this subject will surely think that it deserves a look. 

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Mark Hunacek (mhunacek@iastate.edu) is a Teaching Professor Emeritus at Iowa State University.