Linear Algebra and Geometry

PREFACE

PREFACE Linear algebra is now included in the undergraduate curriculum of most universities. It is generally recognized that this branch of algebra, being less abstract and directly motivated by geometry, is easier to understand than someother branches and that because of the wide applications it should be taught as early as possible. The present book is an extension of the lecture notes for a course in algebra and geometry given each year to the first-year undergraduates of mathematics and physical sciencesin the University of Hong Kong since 1961. Except for some rudimentary knowledge in the language of set theory the prerequisites for using the main part of this book do not go beyond Form V1 level. Since it is intended for use by beginners, much care is taken to explain new theories by building up from intuitive ideas and by many illustrative examples, though the general level of presentation is thoroughly axiomatic. The book begins with a chapter on linear spaces over the real and the complex field in leisurely pace. The more general theory of linear spaces over an arbitrary field is not touched upon since no substantial gain can be achieved by its inclusion at this level of instruction . In $3 a more extensive knowledge in set theory is needed for formulating and proving results on infinite-dimensional linear spaces. Readers who are not accustomed to these set-theoretical ideas may omit the entire section. Trying to keep the treatment coordinate-free, the book does not follow the custom of replacing any space by a set of coordinates, and then forgetting about the space as soon as possible. In this spirit linear transformations come (Chapter11)before matrices (Chapter V). While using coordinates students are reminded of the fact that a particular isomorphism is given preference.฀Another feature of the book is the introduction of the language and ideas of category theory ($฀8) through which a deeper understanding of linear algebra can be achieved. This section is written with the more capable students in mind and can be left out by students who are hard pressed for time or averse to a further level of abstraction. Except for a few incidental remarks, the material of this section is not used explicitly in the later chapters. Geometry is a less popular subject than it once was and its omission in the undergraduate curriculum is lamented by many mathematicians . Unlike most books on linear algebra, the present book contains two substantial geometrical chapters (Chapters 111฀and IV) in which affine and projective geometry are developed algebraically and in a coordinate-free manner in terms of the previously developed algebra. I hope this approach to geometry will bring out clearly the interplay of algebraic and geometric ideas. The next two chapters cover more or less the standard material on matrices and determinants. Chapter V11 handles eigenvalues up to the Jordan forms. The last chapter concerns itself with the metric properties of euclidean spaces and unitary spaces together with their linear transformations. The author acknowledges with great pleasure his gratitude to Dr D.L.C Chen who used the earlier lecture notes in her classes and made several useful suggestions. I am especially grateful to Dr C.B. Spencer who read the entire manuscript and made valuable suggestion for its improvement both mathematically and stylistically. Finally I thank Miss Kit-Yee So and Mr K.W. Ho for typing the manuscript.฀ K. T. Leung University of Hong Kong January 1972 ...