Finite Dimensional Vector Spaces: University Series in Undergraduate Mathematics Paperback – September 14, 2013

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.

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 linear algebra course using this book then there are a few deficiencies I would try to correct, however.

1. The polar decomposition is covered but the singular value decomposition (for linear transformations between different inner product spaces) is not mentioned. This is a pretty big gap in terms of applications, although it's trivial 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. I remember being quite confused about what a tensor product was until I realized some years later that the tensor product of L^2(R) with itself is L^2(R^2), an example which I think should be mentioned, even if it requires a bit of vagueness about Lebesgue measurability.

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 polynomials of A.

The functional calculus could be introduced with the following

Problem: For a function f:R->R and a self-adjoint matrix A (or more generally a possibly nonunitarily diagonalizable matrix A with complex eigenvalues and f:C->C) define f(A)=P(A) where P is a polynomial chosen so that P(lamda) = f(lamda) for all lamda.

A. Show that f(A) is well defined.
B. Show that (f+g) (A) =f(A)+g(A), (fg)(A) = f(A) g(A), and (f composed g) (A) = f(g(A))
C. Show that d/dt exp(tA) = Aexp(tA)
D. Show that it is false that d/dt exp(A+tB) = B exp(A+tB) for matrices A and B.
E. If f is differentiable show that d/dt Tr f(A+tB) = Tr B f'(A+tB)
F. Show that the rotation U(theta) of R^2 by an angle theta is given by Exp(tJ), where J={{0,1},{-1,0}}.

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.

6. A bit more connection to calculus should be made, if only in optional exercises. Students should know how to compute d/dt det(A) and d/dt A^(-1), where A=A(t) is a matrix-valued function of time t.

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!