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Advanced Linear Algebra (Graduate Texts in Mathematics) Softcover reprint of hardcover 3rd ed. 2008 Edition

4.8 out of 5 stars 6 customer reviews
ISBN-13: 978-1441924988
ISBN-10: 1441924981
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Editorial Reviews

Review

From the reviews of the first edition:

"… The book is very well written and has a good set of exercises. It is a suitable choice as a graduate textbook as well as a reference book." A.A. Jafarian for ZentralblattMATH

From the reviews of the second edition:

"In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials. … As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields. … the exercises are rewritten and expanded. … Overall, I found the book a very useful one. … It is a suitable choice as a graduate text or as a reference book." (Ali-Akbar Jafarian, Zentralblatt MATH, Vol. 1085, 2006)

"This is a formidable volume, a compendium of linear algebra theory, classical and modern … . The development of the subject is elegant … . The proofs are neat … . The exercise sets are good, with occasional hints given for the solution of trickier problems. … It represents linear algebra and does so comprehensively." (Henry Ricardo, MathDL, May, 2005)

From the reviews of the third edition:

“This is the 3rd edition of a well written graduate book on linear algebra. … The list of references has been enlarged considerably. The book is suitable for a second course on linear algebra and/or a graduate text, as well as a reference text.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, May, 2014)

"This is the 3rd edition of a well written graduate book on linear algebra. … The book covers a wide range of topics in a moderate length and careful manner. … The list of references has been enlarged considerably. … is suitable for a second course on linear algebra and/or a graduate text, as well as a reference text." (A. Arvanitoyeorgos, Zentralblatt MATH, Vol. 1132 (10), 2008)

From the Back Cover

For the third edition, the author has added a new chapter on associative algebras that includes the well known characterizations of the finite-dimensional division algebras over the real field (a theorem of Frobenius) and over a finite field (Wedderburn's theorem); polished and refined some arguments (such as the discussion of reflexivity, the rational canonical form, best approximations and the definitions of tensor products); upgraded some proofs that were originally done only for finite-dimensional/rank cases; added new theorems, including the spectral mapping theorem; considerably expanded the reference section with over a hundred references to books on linear algebra.

From the reviews of the second edition:

"In this 2nd edition, the author has rewritten the entire book and has added more than 100 pages of new materials....As in the previous edition, the text is well written and gives a thorough discussion of many topics of linear algebra and related fields...the exercises are rewritten and expanded....Overall, I found the book a very useful one....It is a suitable choice as a graduate text or as a reference book."

Ali-Akbar Jafarian, ZentralblattMATH

"This is a formidable volume, a compendium of linear algebra theory, classical and modern... The development of the subject is elegant...The proofs are neat...The exercise sets are good, with occasional hints given for the solution of trickier problems...It represents linear algebra and does so comprehensively."

Henry Ricardo, MAA Online

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Product Details

  • Series: Graduate Texts in Mathematics (Book 135)
  • Paperback: 526 pages
  • Publisher: Springer; Softcover reprint of hardcover 3rd ed. 2008 edition (November 19, 2010)
  • Language: English
  • ISBN-10: 1441924981
  • ISBN-13: 978-1441924988
  • Product Dimensions: 6 x 1.2 x 9 inches
  • Shipping Weight: 2 pounds (View shipping rates and policies)
  • Average Customer Review: 4.8 out of 5 stars  See all reviews (6 customer reviews)
  • Amazon Best Sellers Rank: #1,484,577 in Books (See Top 100 in Books)

Customer Reviews

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Top Customer Reviews

Format: Hardcover
Linear algebra is crucial to anyone in a mathematical or technical field. To the pure or applied mathematician, it is the bread and butter -- a lot of fundamental theorems (even in quite advanced fields like algebraic geometry) ultimately come down to a calculation using linear algebra.

In any case, this book is brilliant for the moderately advanced student who knows the basics (maybe sketchily) and wants an extremely comprehensive, rigorous, and coherent review and reordering of his or her linear algebra knowledge. I knew most of the topics in this book in a superficial way, but reading it is quite fulfilling because it all comes together at once.

The choice of topics and the angles from which they are presented is extremely strong. The Jordan and Rational Canonical Forms get a full and rigorous treatment. Unlike many linear algebra books, which use some ugly matrix-related kludge in the proofs of the classification theorems, this book does these topics from the algebraic perspective (i.e., as decompositions of modules over principal ideal domains). Inner product spaces are done in their own substantial chunk of the book, where all the essential ideas are developed abstractly and well. Sometimes linear algebra books focus too much on particular examples of inner product spaces or resort to "magical" proofs of important inequalities. This book takes care to build up important lemmas so that big results fall out "naturally". It is by far the best abstract treatment of inner products that I have read (although it should be supplemented by a knowledge of some of the standard examples, which can be found in a typical introductory textbook).

The proofs are the most elegant possible, with no ugliness or nonsense.
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Format: Hardcover
Mathematics books are often considerably more difficult to read than their authors prepare their audiences to believe; this book is a happy exception. It is written for an audience of readers at a specific place in their studies (ones who know linear algebra but want to take their understanding of it to a deeper level), and it reaches this audience very well. The emphasis of this book is on linear algebra in abstract mathematics; it is less useful for people interested in numerical linear algebra.

As the name suggests, this book requires a fair amount of background. The introductory chapter moves very fast, but is thorough, and exciting to read. The rest of the book presents advanced topics at a more leisurely pace, while still remaining fairly concise. Some difficult concepts, such as the universal property, are introduced several times at several different places in the book, so that someone working through the book will be more familiar with them when it is finally necessary to understand them on a deeper level.

I find the material on modules outstanding; the author explores the analogies between modules and vector spaces, rigorously exploring which analogies hold, and giving examples of cases in which other analogies fail. The presentation of modules in this book differs greatly from that encountered in most abstract algebra texts: while most books focus on modules' similarities to rings and applications in commutative algebra, this text focuses on their similarities to vector spaces and applications to the study linear operators on vector spaces.

One should not be scared by the word "advanced" in the book's title. Although the book covers advanced topics, it is very clear.
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Format: Paperback Verified Purchase
This is a great linear algebra book for anyone who has already had an undergraduate course in linear algebra. It would help even more if the reader has already had at least one semester of abstract algebra (modern algebra). As for the content of the book, it is very readable and it will go well beyond what you may have learned about linear algebra in the past. The first chapter is on vector spaces whereas in more elementary texts, vector spaces are not treated until later chapters - after learning how to compute the solutions to systems of linear equations using Gaussian elimination or LU decomposition or other methods. This book is not about computation but about the development of theory. Bottom line: great book for someone looking to get heavy into the theory of linear algebra and to get a deep understanding of modules, linear operators and transformations, Hilbert spaces, metric spaces, etc. Have fun with it!
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