Introduction to Linear Algebra [antikvár]

PREFACE This book is intended to be an introduction in the geiiuine sense of the word. It is written for first year students majoring in mathematics, science, or engineering who have no background beyond their high-school knowledge. It can be used as a textbook for a one-semester course of linear algebra running parallel to a course of mathematical analysis in which real numbers are treated. There are several fine books on linear algebra which begin with the axioms defining groups, fields, linear spaces, etc., and continue by proving the main theorems of these algebraic structures. There seemed no need for another text of the same kind, but we thought that the approach leading the beginner to the axioms themselves was much neglected. The aim of the lectures of the second author, and now of this book, is to present the development of linear algebra from experience—that is, to bridge the concrete structures (partly taught in high school) and the axiomatic treatment of the subject. With this aim we embarked on the dangerous path of applying intuitive concepts at the beginning of a rigorous text. Accordingly, in Chapter I we deal with common vector algebra in the ordinary space. Here vectors are models of the intuitive concept of displacement. Apart from this, the treatment is rigorous from the start. The subject is arranged as a model for the abstract ideas that are introduced later. Similar reasons explain the inclusion of complex numbers in Chapter II, with emphasis on the linear space structure in the complex field. A fairly comprehensive elementary matrix algebra is given in Chapter III including a straightforward theory of determinants. One more concrete model of linear space is treated here: the set of ordered %-tuples. (The authors believe that this particular structure must be studied before the abstract treatment of linear spaces, to make coordinate representation of vectors possible without confusion.) Chapter IV deals with the theory of systems of linear equations. A survey of the structure of the solution set is given here and we hope that this will be of use to the reader studying, for instance, linear differential and integral equations. The axiomatic definition of the linear space is given in Chapter V. The concept and the properties of the Euclidean space are also treated here. Chapter VI concerns linear operators mapping the linear or the Euclidean space