Customer Reviews

Finite-Dimensional Vector Spaces

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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.

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.

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.

This was one of the two textbooks (along with Rudin's Principles of Mathematical Analysis) that was used for the hot-shot freshman Math 218x course taught by Elias Stein at Princeton some years ago.

It is a great book, one of my all-time favorites. It requires a bit of mathematical maturity, that is a love of mathematical proof and simplifying abstractions. This book abstractly defines vector spaces and linear transformations between them without immediately introducing coordinates. This approach is vastly superior to immediately extorting the reader to study the algebraic and arithmetic properties n-tuples of numbers (vectors) and matrices (n x n tables of numbers) which parameterize the underlying abstract vectors and linear transformations, respectively.

If I taught a serious linear algebra course using this book then there are a few deficiencies I would try to correct:

1. The polar decomposition is covered but the singular value decomposition (for linear transformations between different inner product spaces) is omitted. This is a pretty big gap in terms of applications, although it's easy to get the singular value decomposition if you have the polar decomposition.

2. The identification of an reflexive vector space with its double-dual was a stumbling block for me when I took the course. There was no mathematical definition of "identify", and so I was confused. Perhaps a good way to remedy this is to give a problem with the example of the Banach space L^p (perhaps just on a finite set of just two elements), and show how L^p is dual to L^p'.

3. The section on tensor products should be improved and expanded, especially in light of the new field of quantum information theory.

4. It would be nice to have a problem (or take-home final) where the reader proves the spectral theorem using minimal polynomials without recourse to determinants, and introduces the functional calculus just using polynomials. It is disturbing to see how many physics grad students are so hung up thinking of eigenvalues only as roots of the characteristic polynomial that they can't understand properties of the spectrum of a self-adjoint transformation A by considering polynmomials of A.

5. I missed the connection between polynomials of a matrix and the Jordan form when I learned linear algebra from this book. Perhaps the following problems would be helpful, and give a proper finite-dimensional introduction to the Dunford calculus (before it is slightly-obfuscated in infinite dimensions using Cauchy's formula):

Problem A: Let P be a complex polynomial, and let A be a linear transformation on a complex vector space, with eigenvalues {z_1,...,z_n}, and let the Jordan block corresponding to z_k have a string of 1's that is at most s_k elements long. Then the value of P(A) is determined by the values of P and its first s_k derivatives at the z_k. (One defines the derivative of a function from C to C by taking a limit of difference quotients, in the same way one defines a derivative of a real function. In particular, the usual rules for differentiating polynomials apply.)

Problem B: (Finite-dimensional Dunford calculus, assuming differentiablity only on the spectrum) Suppose that f:C->C has s_k complex derivatives at the z_k. Define f(A)=P(A), where P is a polynomial with derivatives up to order s_k agreeing with those of f at the z_k. Show that such polynomials always exist. (In particular, f(A) is well-defined by problem A.) Show that (f+g)(A)=f(A)+g(A), f(A)g(A)=(fg)(A), and f(g(A))=h(A), where h is the composition of f and g as functions from C->C.

Problem C: Use B to show that every nonsigular matrix has a square root, as do singular matrices with no 1's in the jordan block for the eigenvalue 0.

Problem D: Are the only matrices with square roots given by problem C?

Except for property (3) above, this is a good book for students who are interested in taking a quantum mechanics or quantum computing course in the future.

If you read this book and like it, then in the future you might want the following graduate-level textbooks:

Bhatia's book "Matrix Analysis".

Reed and Simon's "Methods of Mathematical Physics", especially volume 1 on functional analysis. (This is the infinite-dimensional version of Halmos's book.)

Halmos's "A Hilbert Space Problem Book"

You'll certainly need to learn some analysis before tackling the last two books, though!

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.

Halmos uses informal language and is not afraid to let his feelings show from time to time. There are a couple of exasperated remarks about the formalism that I find very funny. Because of this personal quality of the prose I feel affection for the work that I don't feel for many textbooks.

For me this book was just fine as the introductory text. But I was a physics student and must have had from that a working knowledge of vectors and transforms in orthogonal spaces, and I knew and liked the rigorous approach to math.

I still feel I would not like to be without the book.

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.

This book is very concise and covers everything you learned in undergraduate Linear Algebra course and much more. Pre-reqs I would say have to be a solid prep in Linear Algebra and knowing how to do proofs (a course or two in Real Analysis wouldn't be a bad idea). The book contains a handful of examples and exercises but most of the exercises are proofs. Recommended for seniors and graduate students only.

This book is correctly regarded by many mathematicians as a definitive introductory text on its subject. Be aware that it is not well suited as YOUR first text. If you have never dealt with finite dimensional vector spaces and are, typically, an undergrad, then a more standard and longer text will be an easier read.

But, once you have some knowledge of vector spaces, it pays to read Halmos. He provides a succinct exposition that explains the main ideas in an engaging and understandable style. Standard texts, directed towards a broader maths audience, often require more verbosity.

The only drawback with Halmos' book is the relative paucity of problems. However, if you have another text, that should typically provide you with the necessary problem sets.

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.