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Finite-Dimensional Vector Spaces

Paul R. Halmos
Springer Verlag
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
[Reviewed by
Allen Stenger
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This is a classic but still useful introduction to modern linear algebra. It is primarily about linear transformations, and despite the title most of the theorems and proofs work for arbitrary vector spaces. The presentation doesn’t seem dated at all, except for the use of the terms proper value and proper vector for eigenvalue and eigenvector (these weren’t standardized when the book was written). A “preliminary” version of the book was published in 1942, and this edition is a reprint of the 1958 Van Nostrand second edition.

My describing this as an “introduction” needs some qualification. The book does start from the beginning and assumes no prior knowledge of the subject. It’s also extremely well-written and logical, with short and elegant proofs. But it has no pictures, no worked examples, no discussion of algorithms or numerical methods, and very little motivation. It’s really hard to know where you’re going if you’ve never studied linear algebra before, so it’s extremely helpful to have already had a course in matrices and linear algebra in \(\mathbb{R}^n\). Most of the book is structured as “define something; prove a few theorems about the something; repeat with a new something.” After 55 pages of this we find the dismaying admission “We come now to the objects that really make vector spaces interesting.”

The book is not completely matrix-free, but matrices are definitely a sideline. However, there are numerous matrix examples in the exercises. The book develops a few special matrices, such as upper-triangular, diagonal, and the Jordan normal form, but these come out of analyzing the structure of linear transformations rather than by looking at how matrices are used. Determinants are also developed, but from alternating multilinear forms rather than from systems of linear equations. In fact, I didn’t spot any mention of linear equations in the book.

The exercises are very good, and are a mixture of proof questions and concrete examples. The book ends with a few applications to analysis (Halmos’s true interest in the subject) and a brief summary of what is needed to extend this theory to Hilbert spaces.

I would not recommend the present book for a first course in linear algebra for any audience, not even math majors, but it is a good choice for a second course. Linear algebra is so important today that I think it is better for everyone to start with a concrete approach that uses lots of matrices and calculations. Strang’s Introduction to Linear Algebra is a very good choice, because it is concrete but also takes a linear spaces viewpoint throughout.

Many years after the present book was published Halmos wrote A Linear Algebra Problem Book, that was intended as a supplement to the present volume, but also competes with it because it gives a complete course (inquiry-based) in linear algebra. There’s not much overlap between the two books’ problems, and the latter book also includes hints and complete solutions. I find the later book to be much more accessible than the earlier one, but it takes some daring to teach a course based solely on a problem book. Another good alternative for a second course is Axler’s Linear Algebra Done Right. Its noteworthy feature is that it avoids determinants, but it does have lots of matrices. It is written deliberately as a second course. I think it is a better textbook than either of the Halmos volumes, because it has lots of pictures and examples, although it does not go as deep as the Halmos books do.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

1. Spaces; 2. Transformations; 3. Orthogonality; 4. Analysis; Appendix; Recommended Reading; Index of Terms; Index of Symbols