Advanced Linear Algebra (Graduate Texts in Mathematics, Vol. 135) 3rd Edition by Steven Roman (PDF)

Description

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.

User’s Reviews

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 ZentralblattMATHFrom 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 About the Author Dr. Roman has authored 32 books, including a number of books on mathematics, such as Introduction to the Finance of Mathematics, Coding and Information Theory, and Field Theory, published by Springer-Verlag. He has also written Modules in Mathematics, a series of 15 small books designed for the general college-level liberal arts student. Besides his books for O’Reilly, Dr. Roman has written two other computer books, both published by Springer-Verlag. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The author is an excellent teacher. The content is advanced and very easy to follow up. I self-study to chapters 1-10. The book did a good job to connect the concept of undergraduate-level linear algebra( The structure of Jordan form and so on) and the more abstract level of linear algebra(Modules over a Principal Ideal Domain).Highly recommend

⭐With all due certainty, this text shall be well worth your time and money. It has plenty of extra materials per section. Moreover, you can increase your field of study via extra internet sites. Definitely, this title is for the advanced graduate academic.

⭐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!

⭐Very comprehensive and rigorous; it’s like a linear algebra encyclopedia. I also particularly liked the chapter 0.

⭐The item came in good time and in good condition. It was, however, a copy for the Chinese market which had not been mentioned in the descriptive blurb. It appears to be complete, just printed on thinner paper with a thinner cover and smaller in size than the American version. Would have preferred a non-Chinese printing.

⭐This book is excellent. For instance, compared to “Linear Algebra Done Right” by Sheldon Axler, another very good book, I find this book more advanced, more abstract, and at the same time easier to read.

⭐This is a great book. It has great rxplanations and follows with proofs .

⭐Well written & good explanations

⭐This is an excellent book for consolidating and expanding upon topics covered in the first few years of an undergraduate degree. It begins by revisiting material from introductory linear algebra, but does so in a much more mature “abstract algebra” setting which acts as a pleasant stepping stone into the more graduate-level style. There is a decent survey of module theory which is often neglected or missing from undergraduate texts and It also covers a lot of concepts which I had some vague notion of but hadn’t found the right path to investigate; tensor products, graded algebras, exterior product and symplectic and affine geometries, to name a few. Much more than that I really can’t speak highly enough of the author’s pedagogical skill. He imparts information in a clear concise way, but in such a way that everything comes across as natural and obvious. Opening a section of the book at random may look daunting at first, but you won’t have to read more than a few sentences before the ideas emerge in sharp relief and you genuinely feel like you’ve understood something significant. The best way I can describe it is it’s as though he has explained something the way I would have explained it to myself had I already understood it. Maybe I just have a particular affinity with the style, but I suspect it is more universal than that. All in all I would highly recommend the text to anyone looking to recast their undergraduate material on a stronger footing, either in preparation for further study or even as a summer read before beginning their third or fourth year.

⭐Very good book. Roughly equivalent to a version John Conway’s book on Functional Analysis without the Topology or Analysis.

⭐Excellent

⭐This is the second edition of Steven Roman’s text on linear algebra. The first edition was very good but the typesetting was somewhat amateurish. This edition corrects that problem beautifully and offers a much expanded range of topics. The material is well motivated, the proofs are detailed, and the writing is clear. It is ideal for self-study for somone with a background in introductory linear algebra who wishes to delve more deeply into advanced topics.So why only 3 stars? Answer: after minimal use the binding cracked, and it will only be a matter of time before pages start falling out. Springer should be ashamed at distributing such a poorly bound book. I’ve purchased a number of Springer books lately, and this is not the only time I’ve been disappointed at poor production quality. If I were Steve Roman, I would be upset that my otherwise excellent text is being sold with such an inferior binding (at least in my copy). If you purchase this book – and you should – check immediately that the binding is satisfactory.

⭐The perfect bridge between linear algebra and module theory

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