Tuesday, May 21, 2013

Matrix Groups for Undergraduates by Kristopher Tapp

Matrix Groups for Undergraduates PDF Download Ebook. Kristopher Tapp offers concrete and example-driven explanation 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.

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.

The volume is suitable for graduate students and researchers interested in group theory. It 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.

There is considerable emphasis throughout on examples. Since the classical examples of matrix groups dominate the theory anyway, the focus on examples really has no downside. Throughout the book the author is careful to treat the reals R, the complexes C, and the quaternions H as uniformly as possible. Thus he is able to introduce orthogonal groups O(n), unitary groups U(n), and symplectic groups Sp(n) in a parallel way. Theorem 9.31, appropriately stated without proof, forms a satisfying conclusion to the course. It tells readers that the three sequences they have been intensively studying, together with five exceptional groups they have not seen, form the building blocks of all compact matrix groups.

There is further emphasis on low dimensions and visibility. The first chapter explains in intuitive terms why the group SO(3) of rotations of a globe is three-dimensional. It previews the idea of maximal torus by explaining, "Rotating the globe around the axis through the North and South Pole provides a 'circle's worth' of elements of SO(3)." There are helpful pictures throughout the book. Even in the second-to-last chapter, the low-dimensional isomorphism from Sp(1) to SU(2) and the double cover from SU(2) to SO(3) play a prominent role.

There is review appropriate to the intended readers. Aspects of linear algebra over R and Care reviewed in the process of presenting new material corresponding to H. Chapters 4 and 7 are almost entirely devoted to background material, on point-set topology and manifolds respectively; in each case, everything takes place in ambient Euclidean spaces Rn. Even in the last chapter, a theme is that diagonalization theorems of linear algebra are being revisited.

Each chapter concludes with approximately 15 exercises. Some are theoretical, like 4.1 through 4.6, each of which has the form "Prove Proposition 4.x." Many reinforce theoretical topics by considering them in examples, such as Exercise 6.5 which asks students to describe all one-parameter subgroups of GL1(C) and draw some in the x-y plane. The many exercises would support a course where students regularly present material in class.

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