Matrix groups for undergraduates. (Q2803841)

The contents of the book under review are summarized on the back of its cover: ``The author \dots describe(s) the basic objects of Lie theory: Lie algebras, matrix exponentiation, Lie brackets, maximal tori, homogeneous spaces, and roots.''NEWLINENEWLINEFields and skew-fields are briefly mentioned. Afterwards, matrices with entries in the real numbers, the complex numbers, or the quaternions are considered. The topology of subgroups of the general linear group is described. A matrix group is defined as a subgroup of the general linear group that is closed in the general linear group. These topics are preparations for a study of Lie theory in which matrices play a prominent role.NEWLINENEWLINEThe author gives an inspiring presentation of the topics presented in this book. In the beginning, the author lists a number of ``examples of how amazingly ubiquitous matrix groups have become in mathematics, physics and other fields''. He mentions calculus, linear algebra, and abstract algebra as prerequisites and analysis as optional. He promises to ``develop these analysis topics from scratch''.NEWLINENEWLINETwo new chapters were added in the second edition of the book.NEWLINENEWLINEIn Chapter 10 the author concentrates on the construction of new examples of manifolds; he also introduces the concepts of projective spaces and of Lie groups.NEWLINENEWLINEChapter 11 is devoted to roots and dual roots, the bracket of two root spaces, and the Weyl group.NEWLINENEWLINE For the first edition see [Zbl 1089.20001].