Linear Algebra and Group Theory

This is a very traditional, not to say old-fashioned, text in linear algebra and group theory, slanted very much towards physics. The present volume is a 2011 unaltered reprint of the 1961 McGraw-Hill edition, which was in turn extracted, translated, and edited from Smirnov’s 6-volume Russian-language work by Richard A. Silverman. The Russian work had no exercises, and Silverman added about 400 exercises from a variety of sources. Most of these have brief hints and answers in the back of the book.

The linear algebra portion focuses on matrices and their manipulation and use for solving systems of linear equations. This includes some discussion of vectors and tensors and of covariant and contravariant transformations. The book ignores numerical issues, and gives only Cramer’s rule for solving systems of equations. There is also a chapter on infinite-dimensional spaces, dealing with the Hilbert spaces l² and L². The treatment is very concrete and these spaces are treated only as collections of sequences or functions, not as complete normed vector spaces.

The group theory portion takes the last one-third of the book. It is not the abstract group theory we see in courses today, but is a detailed look at several particular types of concrete groups, in particular rotation and symmetry groups. There is also a fairly long chapter on group representations, that builds on the earlier work on matrices.

The teaching of linear algebra has received much attention in the curriculum in recent years, and I believe that today we have much better linear algebra books than this one. I especially like Strang’s Introduction to Linear Algebra. The treatment of group theory has evolved less, and this book is still useful as a brief introduction to groups in physics. There are a number of more recent and more detailed books on portions of this subject, in various degrees of abstraction. The comprehensive 1962 text Group Theory and its Application to Physical Problems by Morton Hamermesh (Addison-Wesley, reprinted in 1989 by Dover) covers the same ground as the last third of Smirnov, but in much greater depth.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.