Vectors, Pure and Applied: A General Introduction to Linear Algebra

What can a new book on linear algebra offer when there are already several good introductions available? Thomas Körner says of his book, as if in answer: “There exist several fine books on vectors which achieve concision by only looking a vectors from a single point of view, be it that of algebra, analysis, physics or numerical analysis … This book is written in the belief that it is helpful for the future mathematician to see all these points of view.” Körner’s previous books include The Pleasures of Counting, Fourier Analysis, and A Companion to Analysis: A Second First and First Second Course in Analysis, which by themselves offer excellent grounds to give this book a good look.

The book begins at a basic level: solving systems with two linear equations in two unknowns. Körner suggests that readers skip quickly through familiar material and stop only when they encounter something new. But it is clear that he intends to establish a sound basis for students and so begins at the beginning, in a section called “Two hundred years of algebra”. From that very elementary start he moves on quickly to matrix algebra, determinants, abstract vector spaces and linear maps.

The book is in two parts. The first part (Familiar Vector Spaces) starts in two and three dimensions, includes some geometry, and generalizes first to Rⁿ, then to Cⁿ. Most of the traditional content of an introductory linear algebra course is in this part. A revealing piece of Körner’s approach is the advice he gives to students: “linear maps for understanding, matrices for computation”.

The second part (General Vector Spaces) has more advanced material and a more abstract treatment. Körner starts by introducing the space of linear maps from a vector space to itself, and follows with a chapter on polynomials in that space (on the way to establishing the Cayley-Hamilton theorem). Later he introduces vector spaces over other fields as “vector spaces without distance” and infinite-dimensional vector spaces as “spaces with distance”. These, he tells the reader, are the two natural ways to generalize finite dimensional vector spaces over R or C. He concludes with a chapter on quadratic forms.

The title says “pure and applied” but the book is lean on the application side. The applications I could identify include: a discussion early on of operations counts for matrix computations, a short section on Shamir’s “secret sharing” algorithm in cryptography, a couple of chapters on tensors with some indications of how they’re used in physics, and a section on error correcting codes.

The book has an abundance of exercises. They come in two varieties. Many exercises are integrated with the text and intended to help the student work with material that’s just been presented; these are straightforward and are designed to be solved or just read. The end-of-chapter exercises are generally more challenging and are aimed at helping the student integrate results presented in the previous chapter.

Körner as a writer is crisp, clear and witty. His books are a joy to read. This one is a solid introduction to linear algebra with plenty of material beyond that for enrichment and further study. The title suggests more in the way of applications than the book supplies, but could be easily supplemented from other sources.

Bill Satzer (wjsatzer@mmm.com) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.