- 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: 131 customer reviews
- Amazon Best Sellers Rank: #552,167 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
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But the material is mostly theoretically interesting, and does not cover many of the computational tricks in a normal linear algebra class — Gauss-Jordan elimination, a hugely important topic, is not even mentioned. There are so many other things missing, like calculations. You work more with operators and vector spaces than with matrices, and finishing this book won't help you understand the matrix terminology that's common in linear algebra. The book is more suited as a primer to a higher-level theoretical class, like operator theory, functional analysis, or modern algebra. It cannot be a prerequisite to practical/applied courses, like, say, statistics or machine learning. I feel like I've learned a lot after finishing the book, but I don't feel prepared for courses that require a 'working knowledge' of linear algebra.
If you're at all interested in theoretical aspects of algebra or being gently introduced to good proofs, this book will appeal to you. I had never done theoretical math before, and this book was interesting and accessible.
Their education is the responsibility of us all, and how often we forget the old ways...
The proofs are clear but do require the reader to fill in some gaps. This is intended. Open to any page and witness the clarity that so often escape the best efforts of a certain class of instructor; Axler will not suffer any unmotivated concepts. Everything builds from previous definitions until there's just enough structure to flesh out the chapter objectives, thus there's little fat to distract the reader.
Moreover, Axler is so badass that he does away with determinants until the last chapter of the book, he's so pimp he just didn't need any stinking determinants in his proofs. That's right, the last chapter introduces trace and determinants and proceeds to bring everything together into a magnificent mic drop.
I now finally understand why determinants are inextricably tied to notions of volume, and why we must multiply by the Jacobian when performing change of variables in multi-variable integrals, and so on.
A newcomer to linear algebra will get very little of use here, save for the clearest definitions I've ever seen regarding the structure of vector spaces, subspaces and linear operators. For a more applied/introductory approach to linear algebra, one can do much worse than Strang.
I now feel much more comfortable moving onto a graduate-level Linear algebra course after visiting Axler's book, as such, it will be an invaluable reference moving forward.
If you're an undergraduate who planning on taking any higher level math courses in subjects like numerical analysis, mathematical statistics, or abstract algebra, do yourself a favor and get this book! It will totally help you to understand the underlying principles of concepts like continuous functions on multidimensional spaces, hypothesis testing, and abstract fields. This book will also be extremely helpful if you are going into any variation of computational mathematics where you have to use programming languages like MatLab or R that already assume you have a good working knowledge of finite-dimensional vector spaces (not to be mistaken with standard matrix operations).
On the other hand, if you are just having trouble following along in your standard undergraduate LA class or are looking for a good study aid or textbook supplement I would recommend something more along the lines of Linear Algebra by Strange. A quick way to know if this is you, just ask yourself, "Do the problems I'm trying to solve require the use of determinants?" If the answer is yes, then this is probably not the book for you.
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