Matrix Groups for Undergraduates (Student Mathematical Library) (Student Mathematical Library, 79) by Kristopher Tapp (PDF)

Description

Matrix groups touch an enormous spectrum of the mathematical arena. This textbook brings them into the undergraduate curriculum. It makes an excellent 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 the basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, maximal tori, homogeneous spaces, and roots. This second edition includes two new chapters that allow for an easier transition to the general theory of Lie groups.

User’s Reviews

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

⭐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!

⭐This book is a great introduction to matrix groups and related ideas. The author explains the basic ideas in a clear, concise, and precise way. Although there are many excellent texts on matrix groups and more abstract properties of groups, this book provides the most accessible introduction to the subject that I have found. The book is short and easy to read through, compact, and economically priced. I strongly recommend reading this book before attempting to delve into more advanced texts.The clear and unified treatment of the real, complex, and quaternion groups is *very* nice. Overall, the writing style is so lucid, it is the kind of book where you feel that the writer is teaching you personally, rather than lecturing to an empty hall.Because the book provides such an excellent introduction to the subject, I give it the full 5 stars. The book has a few typos and gaps, but most are pretty obvious. I hope that the author will expand on this book in a future edition, perhaps including a chapter on basic group theory. When you finish reading the book, your only complaint will be that it isn’t longer! Given the excellent exposition, I will be on the lookout for any future texts from this author.

⭐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.

⭐This book gathers the important properties of matrix groups and shows where the studies of these groups tend to. Indeed, the great thing about such groups is that they are :a.) groupsb.) topological groupsc.) smooth manifolds (for most of them)And usually in literature, you don’t find these properties treated in the same book. That would be the original thing about this one.It’s clearly undergraduate level, but well explained. The only regret if I may say, is that many proofs are left as exercise, and I know it must be a size problem with the editor, but it’s always better to have an “adult” view on the proofs instead of doing them on our own (that’s why we have exercises, to use the theorems and properties).

⭐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!

⭐Il libro che ho comprato mi piace ed è utile per i miei studi.El libro que he comprado me gusta y es útil para mis estudios.I like the book I bought; it is useful for my studies.

⭐I hate books with many exercises without solutions.

⭐The book is free online and with much better reproduction of the photographs in the figures. Otherwise, the book is as the other readers have indicated.

⭐Ottimo testo per un corso semestrale.Perfect for a semester course

⭐

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