Finite-Dimensional Vector Spaces (Undergraduate Texts in Mathematics) [German]

by Paul R. Halmos

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• Format: Hardcover
• Language: German
• ISBN: 3540900934
• ISBN13: 9783540900931
• Release Date: August 2000
• Publisher: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
• Weight: 1.17 lbs.

Customer Reviews

A Classic for the mathematically-inclined. Good preparation for learning quantum mechanics.

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

Very clear, (only?) for those who think like mathematicians

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.

A Classic

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.

The great classic of linear algebra

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.