- Series: Undergraduate Texts in Mathematics
- Hardcover: 202 pages
- Publisher: Springer; 1st ed. 1958. Corr. 2nd printing 1993 edition (August 20, 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: 28 customer reviews
- Amazon Best Sellers Rank: #561,669 in Books (See Top 100 in Books)
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Finite-Dimensional Vector Spaces (Undergraduate Texts in Mathematics) 1st ed. 1958. Corr. 2nd printing 1993 Edition
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“This is a classic but still useful introduction to modern linear algebra. It is primarily about linear transformations … . It’s also extremely well-written and logical, with short and elegant proofs. … The exercises are very good, and are a mixture of proof questions and concrete examples. The book ends with a few applications to analysis … and a brief summary of what is needed to extend this theory to Hilbert spaces.” (Allen Stenger, MAA Reviews, maa.org, May, 2016)
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However, the Martino Publishing edition (2012) in question is a reprint of the preliminary 1942 Princeton University Press edition. While of historical value, the typesetting of this text (done in typewriter) makes it a challenge to decipher. For instance, the Script and Fraktur letters are written in by hand!
If you are buying this for a class, this is definitely not the edition you are looking for!
On the other hand, the 2nd edition, reprinted by Benediction Classics (2015), with ~200 exercises and beautifully typeset (by 1958 standards) is a true gem. Linear algebra (despite the almost trivial first impression it might give you) is a difficult and subtle subject. It's easy to miss these things when most introductory classes treat it as if it is synonymous with the study of matrix algebra and determinants. Despite having an introductory linear algebra course (semirigorous with some technical definitions and proofs) and graduate matrix computation course under my belt, I didn't understand the true nature of the subject until I studied from this book (as part of a real analysis course).
The emphasis this text places on the coordinate-free (abstract linear algebra) point of view shows you how a mathematician would think about this elementary and classical subject, in light of its modern generalizations (most notably Banach and Hilbert spaces), which form a large part of functional analysis and the theory of linear operators, in particular. The author's main goal was to draw an analogy between the finite-dimensional theory (the subject of this book) and infinite-dimensional generalizations, of benefit for both novices in terms of their future studies and someone reading the book for review who's already studied the latter.
In spite of these praises, this book is admittedly not the most appropriate exposure to higher-level math, though all the proofs are "elementary". Without some appreciation of its applications (either in a pure or applied math setting), the beginner won't see importance of studying linear transformations and subspaces. The most natural setting for learning this material might be during or just before a course in real analysis of several variables. Only then does the importance of linear maps becomes obvious.
Compared to its modern competitor, Axler's Linear Algebra Done Right, FDVS assumes some degree of mathematical maturity, whereas Axler starts by teaching the reader the basics of writing a proof. The newer editions of Axler cater even more to the complete beginner, with a bunch of colors and pretty pictures. (If you need colors and pretty pictures, pure math is probably isn't for you!) Moreover, Axler has this weird obsession against the determinant, a perfectly legitimate coordinate-independent function of a finite-dimensional linear operator, IMHO. And as stated above, I don't think linear algebra is the best setting to introduce mathematical rigor, since it's not a flashy field: it results often appear trivial, boring, or both to the beginner (even though neither claim is actually justifiable on further study.)
The other textbook of comparable coverage is Hoffman and Kunze, much longer, because it tries to include and give a balanced presentation of both the computational and coordinate-free approaches. However, it only makes sense to use a long book (>400 pages) covering both approaches when you have the luxury of a year-long course, and there is enough time to do justice to both.
tl; dr, for the student with a moderate undergrad abstract math background, FDVS is an enlightening presentation of linear algebra the way a pure mathematician sees it. Though on balance better than its modern counterparts, as a word of warning, some notation and terminology are dated, since the text is almost 60 years old!
It is to LA as Rudin is to Analysis, or Spivak is to "Calculus on Manifolds"
I do love this book for its terseness. The subject is very well described, although I definitely think some subsections have no motivation. Lectures on Linear Algebra, is a good book to supplement for theoretical content.
I do not think this gap is really a fault of the author, than merely a reflection of the very different direction that the teaching undergraduate mathematics has taken since the author published this book. A much more accessible treatment of finite-dimensional spaces is "Linear Algebra Done Right" by Axler. Axler's book really does a much better job of helping a student of mathematics make the transition from "LA for engineers" to the LA of upper division math courses.
Downsides: Terse, very little explanation. This is why I recommend (and use) this book as a reference. No solutions (I prefer it this way, but hints would be nice for some of the exercises).
Overall: The breadth of the material is simply incredible at such a great price. Halmos is a celebrated mathematician whose style is well worth studying. Perfect for advanced study / general linear algebra reference.