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Finite-Dimensional Vector Spaces (Undergraduate Texts in Mathematics) Hardcover – September 1, 1993

ISBN-13: 978-0387900933 ISBN-10: 0387900934 Edition: 1st ed. 1974. 5th printing 1993

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Editorial Reviews

Review

“The theory is systematically developed by the axiomatic method that has, since von Neumann, dominated the general approach to linear functional analysis and that achieves here a high degree of lucidity and clarity. The presentation is never awkward or dry, as it sometimes is in other “modern” textbooks; it is as unconventional as one has come to expect from the author. The book contains about 350 well placed and instructive problems, which cover a considerable part of the subject. All in all this is an excellent work, of equally high value for both student and teacher.” Zentralblatt für Mathematik
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Product Details

  • Series: Undergraduate Texts in Mathematics
  • Hardcover: 202 pages
  • Publisher: Springer; 1st ed. 1974. 5th printing 1993 edition (September 1, 1993)
  • Language: English
  • ISBN-10: 0387900934
  • ISBN-13: 978-0387900933
  • Product Dimensions: 6.1 x 0.5 x 9.2 inches
  • Shipping Weight: 1.1 pounds (View shipping rates and policies)
  • Average Customer Review: 4.0 out of 5 stars  See all reviews (23 customer reviews)
  • Amazon Best Sellers Rank: #966,829 in Books (See Top 100 in Books)

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Customer Reviews

This book is the best if you are looking for an abstract approach to linear algebra.
Manfred Mann
If that _is_ what you want, you may still find this book hard going, but the rewards will be worth the effort.
John S. Ryan
He provides a succinct exposition that explains the main ideas in an engaging and understandable style.
W Boudville

Most Helpful Customer Reviews

64 of 65 people found the following review helpful By John S. Ryan on September 20, 2003
Format: Hardcover
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.
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38 of 39 people found the following review helpful By henrique fleming on May 26, 2000
Format: Hardcover
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.
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22 of 22 people found the following review helpful By Neal Jameson on May 19, 2002
Format: Hardcover
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.
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17 of 17 people found the following review helpful By H Roller on August 11, 2007
Format: Hardcover
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.
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