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Matrix groups are a beautiful subject and are central to many fields in mathematics and physics. They touch upon an enormous spectrum within the mathematical arena. This textbook brings them into the undergraduate curriculum. It is excellent for a one-semester course for students familiar with linear and abstract algebra and prepares them for a graduate course on Lie groups. Matrix Groups for Undergraduates is concrete and example-driven, with geometric motivation and rigorous proofs. The story begins and ends with the rotations of a globe. In between, the author combines rigor and intuition to describe basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, and maximal tori. The volume is suitable for graduate students and researchers interested in group theory.
- Sales Rank: #1608810 in Books
- Brand: Brand: American Mathematical Society
- Published on: 2005-06-13
- Original language: English
- Number of items: 1
- Dimensions: 8.25" h x 5.50" w x .50" l, .50 pounds
- Binding: Paperback
- 166 pages
- Used Book in Good Condition
Review
"this is an excellent, well-written textbook which is strongly recommended to a wide audience of readers interested in mathematics and its applications. The book is suitable for a one-semester undergraduate lecture course in matrix groups, and would also be useful supplementary reading for more general group theory courses." ---- Mathematical Reviews
Most helpful customer reviews
1 of 1 people found the following review helpful.
As for reviewers who were disappointed that the author chose not to provide proofs for ...
By AmazonDan
Very well written book. This book provides many more insights into groups than any other book that I have read. After reading it I was able to return to more advance texts and get more out of them. As for reviewers who were disappointed that the author chose not to provide proofs for everything I think this is not a fair attitude. The fact is that their are plenty of books out there that fulfill that particular need. There are not so many books that go beyond that and provide insight and connections that are difficult to attain early in your studies. I do not see the need to provide a proof if a theorem is 1)understandable and 2) it is easily motivated by other means. The author is definitely in tune with his readers (most of them I think). I will be investigating the other books by this author.
10 of 11 people found the following review helpful.
Phenomenal intro to Lie groups / algebras via matrix groups suitable for self-study
By gengogakusha
It's true as the other reviewers have said: this is an exceptionally good introduction to Lie groups and Lie algebras via matrix groups. It's also suitable for self-study provided you have the required math background. Although it's very much in the definition-theorem-proof mode, there's plenty of insightful and well-written exposition motivating the formal development.
My only complaint is that there are no solutions to any exercises and even worse, the proofs of a number of propositions are left to the reader. This detracts from the value of the book for self-study, but not too much because on balance, in my view, the sheer excellence of the presentation pretty much makes up for that all too common "sin of omission".
The book is extremely concise: only about 137 small pages excluding exercises (no solutions) and back matter. To illustrate: in the chapter (pp. 5-20), one rapidly covers fields and skew-fields (pp. 7-8), quaternions (pp 8-9); the real, complex and quaternion skew-field inclusions (p. 10); matrices as linear transformations (pp. 15-16); general linear groups (pp. 17-18); and finally, change of basis via conjugation (pp. 18-20). In its entirety, the book covers, in the same no-nonsense way: Ch 1. Matrices, Ch. 2 All matrix groups are real matrix groups, Ch. 3 The orthogonal groups, Ch 4. The topology of matrix groups, Ch 5. Lie algebras, Ch 6. Matrix exponentiation, Ch. 7 Matrix groups as manifolds, Ch 8. The Lie bracket, Ch 9. Maximal tori. And, as another reviewer pointed out, the author's AMS website has a free download, Ch 10. Roots, and an errata sheet.
The pace might seem daunting for a first introduction, especially for self-study, but the exposition is so crystal clear that I could hardly put the book down (and I am not a mathematics major, just someone interested in learning the mathematics required for modern physics).
Autodidacts should especially note that it is essential that you have the prerequisites as stated in the introduction on p 7. To wit: multivariable calculus & basic analysis; abstract algebra (groups, subgroups, quotient groups, fields, morphisms, conjugation) and standard linear algebra including linear transformations and relation to matrices, eigenvalues/eigenvectors. You also need to be confident in your abstract mathematical skills; proofs are clear but they're generally quite abstract and concise; there's no hand-holding in that area.
If you're ready for the fast lane, fasten your seat belt and enjoy the ride!
0 of 0 people found the following review helpful.
Definitely the best book in the subject.
By Amazon Customer
Definitely the best presentation of the subject I've ever seen. I am consistently frustrated by introductory textbooks on Lie groups 'n' stuff, because they assume far too much background knowledge. This textbook actually assumes a background only in linear algebra and group theory, and builds everything else from the ground up. It's short and sweet, and wonderfully written. Highly recommended!
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