Finite Dimensional Vector Spaces

Rating:
Summary: Wonderful treatment of linear algebra
Review: (this review referrs to the 1958 edition). This text treats linear algebra much more formally than it is usually treated. Halmos is able to cover a massive amount of material in a short time, though I think the book could have benefitted from a bit more abstraction. He deals primarilly with real vector spaces, and I think that he should have dealt with vector spaces over arbitrary fields. Still, this treatment is far superior to most linear algebra texts, and I advise it to anyone who already has some experience with linear algebra and a fair amount of mathematical maturity.

Linear algebra is one of the most important foundations of mathematics, and Halmos is both a great writer and a significant researcher in operator theory.

Rating:
Summary: Can you say obfuscate?
Review: Concise, yes it is. Overly concise? Absolutely! I was completely, 100% befuddled by the thing! Put it this way, this is not a useful text if you are trying to learn Linear Algebra for the first time. Try ANY other linear algebra book for a first go. Then, maybe later, come back to this.

Rating:
Summary: Very clear, (only?) for those who think like mathematicians
Review: Halmos always exemplifies clarity in writing, but sometimes only for those who either think like mathematicians or are working on learning how to do so. Others should stay away, and stop blaming Halmos if their instructors inappropriately prescribe this book for students for whom it is not suitable.

Rating:
Summary: Linear algebra for mathematicians
Review: I've just been looking on Amazon to see how some of my favorite old math texts are doing. I used this one about twenty years ago as a supplementary reference in a graduate course, and I still have my copy.

Everybody with some mathematical background knows the name of Paul Richard Halmos. I saw him speak at Kent State University while I was an undergraduate there (some twenty-odd years ago); to this day I remember the sheer elegance of his presentation and even recall some of the specific points on which, like a magician, he drew gasps and applause from his audience of mathematicians and math students.

This book displays the same elegance. If you're looking for a book that provides an exposition of linear algebra the way mathematicians think of it, this is it.

This very fact will probably be a stumbling block for some readers. The difficulty is that, in order to appreciate what Halmos is up to here, you have to have _enough_ practice in mathematical thinking to grasp that linear algebra isn't the same thing as matrix algebra.

In your introductory linear algebra course, linear transformations were probably simply identified with matrices. But really (i.e., mathematically), a linear transformation is a special sort of mathematical object, one that can be _represented_ by a matrix (actually by a lot of different matrices) once a coordinate system has been introduced, but one that 'lives' in the spaces with which abstract algebra deals, independently of any choice of coordinates.

In short, don't expect numbers and calculations here. This book is about abstract algebraic structure, not about matrix computations.

If that's not what you're looking for, you'll probably be disappointed in this book. If that _is_ what you want, you may still find this book hard going, but the rewards will be worth the effort.

Rating:
Summary: Am I nuts, or is this a TERRIBLE book?
Review: The book is widely acclaimed, so I don't need to say much about it. Perhaps, the most important fact about the book is that it treats general finite dimensional vector spaces, not the specific cases of R^n and C^n. This liberation helps the reader apply linear algebra techniques to more general scenarios such as finite dimensional function spaces. The exercises use different finite dimensional vector spaces, so that the reader can get a feel for the generality of the methods.

The book is terse at times and requires mathematical maturity (i.e. be familiar with doing rigorous proofs.) I know linear algebra quite well, but I was still left scratching my head a few times wondering about the methods proofs.

I also feel obliged to mention some points in which I think the other reviewers are incorrect:
1. The book always concerns vector spaces over a general field unless Halmos tells you differently, but the exercises generally utilize the real or complex field.
2. The book does not explicitly mention linear mappings between vector spaces of different dimension, but in most scenarios, one can always expand the dimension of the domain or range to make the mapping a mapping between two vector spaces of the same dimension.
3. I would recommend this book as a first book on linear algebra because it will introduce the person to linear algebra without making use of unnecessary coordinate systems that dominate many introductions. Studying matrices and coordinations does very little in helping someone understand the basic theory of linear operators. It only seems to confine their mind to the specific cases of R^n or C^n. The only caveat to first-timers is the book's difficulty.

Rating:
Summary: The great classic of linear algebra
Review: This book has been around for so many years that reviewing it may seem a waste of time. Still, we should not forget that new students keep appearing! Halmos is a wonderful text. Besides the clarity which marks all of his books, this one has a pleasant characteristic: all concepts are patiently motivated (in words!) before becoming part of the formalism. It was written at the time when the author, a distinguished mathematician by himself, was under the spell of John von Neumann, at Princeton. Perhaps related to that is the fact that you find surprising, brilliant proofs of even very well established results ( as, for instance, of the Schwartz inequality). It has a clear slant to Hilbert space, despite the title, and the treatment of orthonormal systems and the spectrum theorem is very good. On the other hand, there is little about linear mappings between vector spaces of different dimensions, which are crucial for differential geometry. But this can be found elsewhere. The problems are useful and, in general, not very difficult. All in all, an important tool for a mathematical education.

Rating:
Summary: overrated, but still good
Review: This book is a good reference, because it contains a lot more material than is contained in most courses, but I don't think I'd want to use it for an intro to linear algebra. It's got stuff that other books don't have, like Hilbert spaces & some analysis stuff in an appendix, tensor products, multilinear forms... It's good as a reference or supplement, but not as a main text, IMO. For an intro, I liked Axler's Linear Algebra Done Right or the Hoffman/Kunze book.

Rating:
Summary: The best abstract linear algebra book out there
Review: This book is the best if you are looking for an abstract approach to linear algebra. It provides elegant proofs to theorems that usually seem long-winded and awkward (like the cauchy-schwarz inequality). Sometimes in your lectures you may get to the point thinking "can't this be proven more elegant?" and you simply open halmos and it is there.

Note that this book does not deal alot with matrices, everything of the theory is there, but you might miss illustrations and applications. In this case I recommend to back it up with Gilbert Strangs Linear Algebra and its Applications, which has an intuitive, matrice-oriented approach.

Considering the price and the wide range of topics often left out in other books (like Nilpotence, Jordanform, Spectral Theorem,...) this simply is the one book you should buy and keep for reference.