- Series: Undergraduate Texts in Mathematics
- Paperback: 251 pages
- Publisher: Springer; 2nd edition (February 26, 2004)
- Language: English
- ISBN-10: 0387982582
- ISBN-13: 978-0387982588
- Product Dimensions: 7.5 x 0.6 x 9.2 inches
- Shipping Weight: 1.2 pounds
- Average Customer Review: 103 customer reviews
- Amazon Best Sellers Rank: #113,802 in Books (See Top 100 in Books)
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Linear Algebra Done Right (Undergraduate Texts in Mathematics) 2nd Edition
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From the reviews:
"This second edition of an almost determinant-free, none the less remarkably far-reaching and didactically masterly undergraduate text on linear algebra has undergone some substantial improvements. First of all, the sections on selfadjoint operators, normal operators, and the spectral theorem have been rewritten, methodically rearranged, and thus evidently simplified. Secondly, the section on orthogonal projections on inner-product spaces has been extended by taking up the application to minimization problems in geometry and analysis. Furthermore, several proofs have been simplified, and incidentally made more general and elegant (e.g., the proof of the trigonalizability of operators on finite-dimensional complex vector spaces, or the proof of the existence of a Jordan normal form for a nilpotent operator). Finally, apart from many other minor improvements and corrections throughout the entire text, several new examples and new exercises have been worked in. However, no mitigation has been granted to determinants. Altogether, with the present second edition of his text, the author has succeeded to make this an even better book."
AMERICAN MATHEMATICAL MONTHLY
"The determinant-free proofs are elegant and intuitive."
"Every discipline of higher mathematics evinces the profound importance of linear algebra in some way, either for the power derived from its techniques or the inspiration offered by its concepts. Axler demotes determinants (usually quite a central technique in the finite dimensional setting, though marginal in infinite dimensions) to a minor role. To so consistently do without determinants constitutes a tour de forces in the service of simplicity and clarity; these are also well served by the general precision of Axler’s prose. Students with a view towards applied mathematics, analysis, or operator theory will be well served. The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library."
"Altogether, the text is a didactic masterpiece."
From the reviews of the second edition:
Linear Algebra Done Right
"The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library."—CHOICE
"A didactic masterpiece."—ZENTRALBLATT MATH
“This book can be thought of as a very pure-math version of linear algebra … . it focuses on linear operators, primarily in finite-dimensional spaces … . Axler has come up with some very slick proofs of things that … makes the book interesting for mathematicians. The book is also very clearly written and fairly leisurely. … Axler concentrates on the properties of linear operators, and doesn’t introduce other concepts unless they’re really necessary.” (Allen Stenger, The Mathematical Association of America, December, 2010)
Top customer reviews
I got this book for a class, and read the second half of the book because I liked it that much. The chapters are fairly short (maybe 10-30 pgs) but instructive. There are about 30 questions at the end of each chapter. The author does a good job of explaining things.
I would recommend this book for anyone who wants a different perspective that isn't immediately applied. This book isn't a "how to use linear algebra to solve your computational problems" book but rather a "mathematical underpinnings of linear algebra through proofs using vector spaces" book. With that said, the proofs aren't super complicated. If you feel you can use linear algebra but don't really *understand* it, then this is definitely a book to consider.
I used this textbook to "replace" the textbook required for my undergraduate Applied Linear Algebra Course. My original "Applied" textbook skipped over necessary abstractions and concepts which I required to actually apply the methodologies effectively.
This text is phenomenal. Not only did it assist me in my applied coursework and functionally replace a much longer and less clear text- but it gave me the motivation to pursue the subject further.
1. Excellent for linear algebra beginners.
2. Excellent clarification of abstract algebraic concepts.
3. Great preparation for abstract mathematics.
4. Excellent exercises.
1. No solutions to exercises.
While it is targetted at the upper undergraduate level of mathematical maturity, I found that it's clear exposition and development of the topics (and the fact that it provides an alternative approach with clear explanations) was useful even in an upper graduate course as a suppliment to the main text. I think it would also do very well as the primary text for an introductory course or for self-study by anyone with reasonable previous exposure to mathematical proofs.
Read the book, and you will forgive him on all counts.
Other reviewers have already been thorough in their praise/criticism of Axler's elegant exposition that deprecates matrices and determinants. The highlight in my view is how Axler cleans up proofs by simplifying notation and carefully abstracting common algorithms into lemmas (like 2.4, his Linear Dependence Lemma) that are used over and over. This greatly improves readability and promotes the development of intuition. Some of his nonstandard choices of notation are used to such great pedagogical effect that they seem to threaten to redefine what is standard. The prose is correspondingly clear, concise, and full of useful motivation for difficult points. The formatting is impeccable - definitions, equations, inequalities, and theorems/lemmas are all given a uniform numbering system, making them easy and unambiguous to cite. Subsidiary comments are relegated to the margins of the book, keeping the main line of exposition free of digressions. The text is quite shockingly free of errors. Finally, the layout has a clean but cheerful flower-power look that reminds the reader that math is about beauty and fun - not just intimidating formalism.
Axler's refusal to refer directly to others for inspiration (he seemingly proudly omits a bibliography) does cause some warts. For instance, when looking at orthogonal projections for optimization, he asks the reader to do inner-product gymnastics in polynomial space on [0,1] instead of on [-1,1]. The latter choice gives rise to the all-important Legendre polynomials, whose symmetry properties are much clearer.
Also, while the pristine algebraic presentation was remarkable, I'd have liked to see more geometric insight in places. I got into this book because my undergraduate linear algebra experience, with Apostol Vol. 2, was so frustrating - all of the sweeping and magical structure theorems of self-adjoint operators and so forth seemed to reduce to incomprehensible index-pushing. For me, what finally cleared up these notions to me was drawing, on graph paper, the fate of vectors in R^2 under various linear operators. This was not in the book, but Axler's inclusion of the theory of polar and singular-value decompositions did give some important tools to help unravel these beautiful but elusive issues. Finally, the crystal clarity of the exposition rolls off in Chapters 8 and 9 when getting into the structure theory of general operators on real and complex vector spaces. The symbols get more abstruse, and the arguments get more murky. But I've never seen another author make anything but a mess of, say, the proof of Jordan form. It is hard stuff, and it is not fair to be too hard on authors for failing to make it look easy.
The end-of-chapter problems are abundant enough to give a good feel for the material, with an appropriate range of difficulties for an advanced undergraduate book. There are enough of the routine computations and simple proofs that familiarize readers with the new machinery they are learning, but at least a proof or two in each chapter require creative constructions to complete. I just finished the last of the 224 problems, a task that took me five years' worth of sporadic effort in my free time and vacations as a high school math teacher and then as a graduate student in chemistry. A few problems took me the better part of a year to figure out, though this was without the benefit of collaboration. I found the equivalent of at one sequence of problems (problems 6-8 in Chapter 6) as a starred problem in a graduate functional analysis text. I consider myself a good but not award-winning math student, so this indicates that the problems are consistently tractable but can get pretty tough in places. Axler does not mark his most difficult problems as such; for the teacher assigning Axler's problems for a course, then, it is imperative to work through the problems beforehand.
All told, this is quite a remarkable book. I now feel like I understand linear algebra, something I couldn't say when I first studied the subject eight years ago. The title does not do it justice.