Summary and Info
Mathematicians frequently use geometrical examples as aids to the studyof more abstract concepts and these examples can be of great interest intheir own right. Yet at the present time little of this is to be found inundergraduate textbooks on mathematics. The main reason seems to bethe standard division of the subject into several watertight compartments,for teaching purposes. The examples get excluded since theirconstruction is normally algebraic while their greatest illustrative valueis in analytic subjects such as advanced calculus or, at a slightly moresophisticated level, topology and differential topology.Experience gained at Liverpool University over the last few years, inteaching the theory of linear (or, more strictly, affine) approximationalong the lines indicated by Prof. J. Dieudonne in his pioneering bookFoundations of Modern Analysis , has shown that an effective coursecan be constructed which contains equal parts of linear algebra andanalysis, with some of the more interesting geometrical examples includedas illustrations. The way is then open to a more detailed treatmentof the geometry as a Final Honours option in the following year.This book is the result. It aims to present a careful account, fromfirst principles, of the main theorems on affine approximation and totreat at the same time, and from several points of view, the geometricalexamples that so often get forgotten.The theory of affine approximation is presented as far as possible in abasis-free form to emphasize its geometrical flavour and its linear algebracontent and, from a purely practical point of view, to keep notations andproofs simple. The geometrical examples include not only projectivespaces and quadrics but also Grassmannians and the orthogonal andunitary groups. Their algebraic treatment is linked not only with athorough treatment of quadratic and hermitian forms but also with anelementary constructive presentation of some little-known, but increasinglyimportant, geometric algebras, the Clifford algebras. On thetopological side they provide natural examples of manifolds and, particularly,smooth manifolds. The various strands of the book are broughttogether in a final section on Lie groups and Lie algebras.