Table Of Content

Linear Algebra and Geometry KAM-TIM LEUNG HONG KONG UNIVERSITY PRESS The Author Dr K.T. Leung took his doctorate in Mathe- matics in 1957 at the University of Zurich. From 1958 to 1960 he taught at Miami University and the University of Cincinnati in the U.S.A. Since 1960 he has been with the University of Hong Kong, where he is now Senior Lecturer in Mathematics and Dean of the Faculty of Science. He is the author (with Dr Doris L.C. Chen) of Elementary set theory, Parts I and II, also published by the Hong Kong University Press. LINEAR ALGEBRA AND GEOMETRY Linear Algebra and Geometry KAM-TIM LEUNG HONG KONG UNIVERSITY PRESS 1974 © Copyright 1974 Hong Kong University Press ISBN 0-85656-111-8 Library of Congress Catalog Card Number 73-89852 Printed in Hong Kong by EVERBEST PRINTING CO., LTD 12-14 Elm Street, Kowloon, Hong Kong 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 some other 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 sciences in 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 VI 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 instruc- tion. 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 (Chapter II) 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. v vl Geometry is a less popular subject than it once was and its omission in the undergraduate curriculum is lamented by many mathe- maticians. Unlike most books on linear algebra, the present book contains two substantial geometrical chapters (Chapters III 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 VII 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 sug- gestion 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 CONTENTS PREFACE ....................................... v .......................... Chapter I LINEAR SPACE I ................. § 1 General Properties of Linear Space 4 A. Abelian groups B. Linear spaces C. Examples D. Exercises ................... §2 Finite-Dimensional Linear Space 17 A. Linear combinations B. Base C. Linear indepen- dence D. Dimension E. Coordinates F. Exercises .................. §3 Infinite-Dimensional Linear Space 32 A. Existence of Base B. Dimension C. Exercises §4 Subspace..................................... 35 A. General properties B. Operations on subspaces C. Direct sum D. Quotient space E. Exercises ............. Chapter II LINEAR TRANSFORMATIONS 45 §5 General Properties of Linear Transformation .......... 45 A. Linear transformation and examples B. Composition C. Isomorphism D. Kernel and image E. Factorization F. Exercises §6 The Linear Space Hom (X, Y) ..................... 62 A. The algebraic structure of Horn (X, Y) B. The associative algebra End (X) C. Direct sum and direct product D. Exercises Dual Space .................................... §7 73 A. General properties of dual space B. Dual transformations C. Natural transformations D. A duality between ..(AX) and E. Exercises §8 The Category of Linear Spaces .................... 84 A. Category B. Functor C. Natural transformation D. Exercises ..................... Chapter III AFFINE GEOMETRY 96 §9 Affine Space .................................. 96 A. Points and vectors B. Barycentre C. Linear varie- ties D. Lines E. Base F. Exercises § 10 Affine Transformations .......................... 113 A. General properties B. The category of affine spaces Chapter IV PROJECTIVE GEOMETRY ................ 118 § 11 Projective Space ................................ 118 A. Points at infinity B. Definition of projective space C. Homogeneous coordinates D. Linear variety E. The theorems of Pappus and Desargues F. Cross ratio G. Linear construction H. The principle of duality I. Exercises ..................... § 12 Mappings of Projective Spaces 141 A. Projective isomorphism B. Projectivities C. Semi- linear transformations D. The projective group E. Exercises ............................. Chapter V MATRICES 155 ..................... § 13 General Properties of Matrices 155 A. Notations B. Addition and scalar multiplication of matrices C. Product of matrices D. Exercises ................ § 14 Matrices and Linear Transformations 166 A. Matrix of a linear transformation B. Square matrices C. Change of bases D. Exercises §15 Systems of Linear Equations ...................... 175 A. The rank of a matrix B. The solutions of a system of linear equations C. Elementary transformations on matrices D. Parametric representation of solutions E. Two interpretations of elementary transformations on matrices F. Exercises Chapter VI MULTILINEAR FORMS .................. 196 § 16 General Properties of Multilinear Mappings ........... 197 A. Bilinear mappings B. Quadratic forms C. Multi- linear forms D. Exercises § 17 Determinants .................................. 206 A. Determinants of order 3 B. Permutations C. De- terminant functions D. Determinants E. Some useful rules F. Cofactors and minors G. Exercises Chapter VII EIGENVALUES ........................ 230 § 18 Polynomials ................................... 230 A. Definitions B. Euclidean algorithm C. Greatest common divisor D. Substitutions E. Exercises § 19 Eigenvalues ................................... 239 A. Invariant subspaces B. Eigenvectors and eigenvalues C. Characteristic polynomials D. Diagonalizable endo- morphisms E. Exercises §20 Jordan Form .................................. 250 A. Triangular form B. Hamilton-Cayley theorem C. Canonical decomposition D. Nilpotent endomor- phisms E. Jordan theorem F. Exercises Chapter VIII INNER PRODUCT SPACES ............... 267 §21 Euclidean Spaces ............................... 268 A. Inner product and norm B. Orthogonality C. SCHWARZ'S inequality D. Normed linear space E. Exercises §22 Linear Transformations of Euclidean Spaces.......... 280 A. The conjugate isomorphism B. The adjoint transformation C. Self-adjoint linear transformations D. Eigenvalues of self-adjoint transformations E. Bi- linear forms on a euclidean space F. Isometry G. Exercises ................................. § 23 Unitary Spaces 297 A. Orthogonality B. The Conjugate isomorphism C. The adjoint D. Self-adjoint transformations E. Iso- metry F. Normal transformation G. Exercises Index........................................... 306