Advanced Linear Algebra (Graduate Texts in Mathematics) [Hardcover]

Steven Roman (Author)

Book Description

Publication Date: October 8, 2007 | ISBN-10: 0387728287 | ISBN-13: 978-0387728285 | Edition: 3rd

This graduate level textbook covers an especially broad range of topics. The book first offers a careful discussion of the basics of linear algebra. It then proceeds to a discussion of modules, emphasizing a comparison with vector spaces, and presents a thorough discussion of inner product spaces, eigenvalues, eigenvectors, and finite dimensional spectral theory, culminating in the finite dimensional spectral theorem for normal operators. The new edition has been revised and contains a chapter on the QR decomposition, singular values and pseudoinverses, and a chapter on convexity, separation and positive solutions to linear systems.

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 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

See all Editorial Reviews

Product Details

• Hardcover: 544 pages
• Publisher: Springer; 3rd edition (October 8, 2007)
• Language: English
• ISBN-10: 0387728287
• ISBN-13: 978-0387728285
• Product Dimensions: 9.4 x 6.5 x 1.2 inches
• Shipping Weight: 1.9 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #934,829 in Books (See Top 100 in Books)

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58 of 60 people found the following review helpful:

5.0 out of 5 stars A real treasure, May 1, 2006

By 

brightSnow (USA) - See all my reviews

This review is from: Advanced Linear Algebra (Graduate Texts in Mathematics) (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. The notation is a gem, without confusing mixes of superscripts and subscripts and nonstandard choices. The exposition is at just the right level (for me at least) -- the steps in proofs that are left as exercises are all reasonable and straightforward, and all the details that are subtle or interesting are filled in, discussed, and emphasized.

I have been looking for a beautiful book on linear algebra of this sort for a long time, and am delighted to have finally found it.

32 of 33 people found the following review helpful:

5.0 out of 5 stars Outstanding clarity; this is a very well-written book, January 18, 2007

By 

Alexander C. Zorach "Alex Zorach" (Lancaster, PA) - See all my reviews
(REAL NAME)   

This review is from: Advanced Linear Algebra (Graduate Texts in Mathematics) (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. When proofs are omitted, it is usually because they are very easy for the reader to supply. The exercises are very valuable (some are critical for understanding the material), but they're not diabolically difficult.

I think this book would make an outstanding textbook for an introductory graduate-level course in linear algebra, or perhaps a senior-level undergraduate course for students with a strong background. It is also very well-suited to self-study. A student with prior background in abstract algebra (group theory, ring theory, etc.) will find this book much more manageable than a student who has not covered such material. People wanting a more introductory text might want to look to the book by Axler, or the old classic by Shilov.

5 of 5 people found the following review helpful:

5.0 out of 5 stars The true "Linear ALGEBRA"., April 16, 2010

By 

Manolis Tsakiris - See all my reviews
(REAL NAME)   

This review is from: Advanced Linear Algebra (Graduate Texts in Mathematics) (Hardcover)

This is linear algebra as rigorous and beautiful as it can be.

The development of the presented concepts is in terms of advanced abstract algebra; particularly

rings and modules. However, no prior knowledge of abstract algebra is required to study the book,

and this is the very element that makes it a masterpiece. Readers unfamiliar though with basic

abstract algebra should expect to progress quite slowly throughout the book, given that they

show devotion and enough faith to let their minds be guided by the author's educating pen.

Diligent study of the book leads to beautiful insights being emerged that are far from trivial. For

this reason, it is highly recommended that the prospective reader possesses a good level of mathematical

maturity and is already familiar to a sufficient degree with linear algebra and matrix analysis, e.g. at

the level of C. Meyer's "Matrix Analysis and Applied Linear Algebra", which is an excellent first study

of the topic.

"Advanced Linear Algebra" is a book that addresses those who love linear algebra (or algebra) and are

serious about mastering the pertaining concepts. The book is invaluable for combining rigor, depth of

exposition, excellent notation and educational character for a topic of immense importance that pervades

almost every aspect of modern quantitative sciences.