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Linear Algebra and Geometry Algebra, Logic and Applications Iand Geometry Gordon and Breach Science Publishers LINEAR ALGEBRA AND GEOMETRY ALGEBRA, LOGIC AND APPLICATIONS A Series edited by R. Gdbel Universitat Gesamthochschule, Essen, FRG A. Macintyre The Mathematical Institute, University of Oxford, UK Volume 1 Linear Algebra and Geometry A. I. Kostrikin and Yu. 1. Manin Additional volumes in preparation Volume 2 Model Theoretic Algebra with particular emphasis on Fields, Rings, Modules Christian U. Jensen and Helmut Lenzing This book is part of a series. The publisher will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details. LINEAR ALGEBRA AND GEOMETRY Paperback Edition Alexei I. Kostrikin Moscow State University, Russia and Yuri I. Manin Max-Planck Institut fur Mathematik, Bonn, Germany Translated from Second Russian Edition by M. E. Alferieff GORDON AND BREACH SCIENCE PUBLISHERS AustraliaCanadaChinaFranceGermanyIndiaJapan LuxembourgMalaysiaThe NetherlandsRussiaSingapore SwitzerlandThailand 0 United Kingdom Copyright © 1997 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands under license by Gordon and Breach Science Publishers. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system, without permission in writing from the publisher Printed in India Amsteldijk 166 1st Floor 1079 LH Amsterdam The Netherlands Originally publishedinRussian in1981asJlrraeiiean Anre6pa m FeoMeTpHR (Lineinaya algebra i geometriya) by Moscow University Press 143uaTenvcTSO Mocxoacxoro YHHBepCHTeTa The Second Russian Edition was published in 1986 by Nauka Publishers, Moscow 143ztaTe.nbcTBO Hayxa Mocxaa © 1981 Moscow University Press. British Library Cataloguing in Publication Data Kostrikiu, A. I. Linear algebra and geometry. - (Algebra, logic and applications ; v. 1) 1. Algebra, Linear2. Geometry 1. TitleII. Manin, IU. I. (Iurii Ivanovich), 1937- 512.5 ISBN 90 5699 049 7 Contents Preface vu Bibliography ix CHAPTER 1 Linear Spaces and Linear Mappings I 1 Linear Spaces 2 Basis and Dimension 3 Linear Mappings 4 Matrices 5 Subspaces and Direct Sums 6Quotient Spaces 7 Duality 8 The Structure of a Linear Mapping 9 The Jordan Normal Form 10 Normed Linear Spaces 11 Functions of Linear Operators 12 Complexification and Decomplexification 13 The Language of Categories 14 The Categorical Properties of Linear Spaces CHAPTER 2 Geometry of Spaces with an Inner Product 92 1 On Geometry 92 2 Inner Products 94 3 Classification Theorems 101 4 The Orthogonalization Algorithm and Orthogonal Polynomials 109 5 Euclidean Spaces 117 6Unitary Spaces 127 7 Orthogonal and Unitary Operators 134 8 Self-Adjoint Operators 138 9 Self-Adjoint Operators in Quantum Mechanics 148 10 The Geometry of Quadratic Forms and the Eigenvalues of Self-Adjoint Operators 156 11 Three-Dimensional Euclidean Space 164 12 Minkowski Space 173 13 Symplectic Space 182 vi A. I. KOSTRIKIN AND Yu. I. MANIN 14 Witt's Theorem and Witt's Group 187 15 Clifford Algebras 190 CHAPTER 3 Affine and Projective Geometry 195 1 Affine Spaces, Affine Mappings, and Affine Coordinates 195 2 Affine Groups 203 3 Affine Subspaces 207 4Convex Polyhedra and Linear Programming 215 5 Affine Quadratic Functions and Quadrics 218 6 Projective Spaces 222 7 Projective Duality and Projective Quadrics 228 8 Projective Groups and Projections 233 9Desargues' and Pappus' Configurations and Classical Projective Geometry 242 10 The Kahier Metric 247 11 Algebraic Varieties and Hilbert Polynomials 249 CHAPTER 4 Multilinear Algebra 258 1 Tensor Products of Linear Spaces 258 2 Canonical Isomorphisms and Linear Mappings of Tensor Products 263 3 The Tensor Algebra of a Linear Space 269 4 Classical Notation 271 5 Symmetric Tensors 276 6 Skew-Symmetric Tensors and the Exterior Algebra of a Linear Space 279 7 Exterior Forms 290 8 Tensor Fields 293 9Tensor Products in Quantum Mechanics 297 Index 303 Preface to the Paperback Edition Courses in linear algebra and geometry are given at practically all univer- sities and plenty of books exist which have been written on the basis of these courses, so one should probably explain the appearance of a new one. In this case our task is facilitated by the fact that we are discussing the paperback edition of our textbook. One can look at the subject matter of this course in many different ways. For a graduate student, linear algebra is the material taught to freshmen. For the professional algebraist, trained in the spirit of Bourbaki, linear algebra is the theory of algebraic structures of a particular form, namely, linear spaces and linear mappings, or, in the more modern style. the theory of linear categories. From a more general viewpoint, linear algebra is the careful study of the mathematical language for expressing one of the most widespread ideas in the natural sciences - the idea of linearity. The most important example of this idea could quite possibly be the principle of linearity of small increments: almost any natural process is linear in small amounts almost everywhere. This principle lies at the foundation of all mathematical analysis and its applications. The vector algebra of three-dimensional physical space, which has historically become the cornerstone of linear algebra, can actually be traced back to the same source: after Einstein, we now know that the physical three-dimensional space is also approximately linear only in the small neighbourhood of an observer. Fortunately, this small neighbourhood is quite large. Twentieth-century physics has markedly and unexpectedly expanded the sphere of application of linear algebra, adding to the principle of linearity of small increments the principle of superposition of state vectors. Roughly speaking, the state space of any quantum system is a linear space over the field of complex numbers. As a result, almost all constructions of complex linear algebra have been transformed into a tool for formulating the fundamental laws of nature: from the theory of linear duality, which explains Bohr's quantum principle of complementarity, to the theory of representations of groups, which underlies Mendeleev's table, the zoology of elementary particles, and even the structure of space-time. vii viii A. I. KOSTRIKIN AND Yu. I . MANIN The selection of the material for this course was determined by our desire not only to present the foundations of the body of knowledge which was essentially completed by the beginning of the twentieth century, but also to give an idea of its applications, which are usually relegated to other disciplines. Traditional teaching dissects the live body of mathematics into isolated organs, whose vitality must be maintained artificially. This particularly concerns the `critical periods' in the history of our science, which are characterised by their attention to the logical structure and detailed study of the foundations. During the last half-century the language and fundamental concepts were reformulated in set-theoretic language; the unity of mathematics came to be viewed largely in terms of the unity of its logical principles. We wanted to reflect in this book, without ignoring the remarkable achievements of this period, the emerging trend towards the synthesis of mathematics as a tool for understanding the outside world. (Unfortunately, we had to ignore the theory of the computational aspects of linear algebra, which has now developed into an independent science.) Based on these considerations, this book, just as in Introduction to Algebra written by Kostrikin, includes not only the material for a course of lectures, but also sections for independent reading, which can be used for seminars. There is no strict division here. Nevertheless, a lecture course should include the basic material in Sections 1-9 of Chapter 1; Sections 2-8 of Chapter 2; Sections 1, 3, 5 and 6 of Chapter 3 and Sections 1 and 3-6 of Chapter 4. By basic material we mean not the proof of difficult theorems (of which there are only a few in linear algebra), but rather the system of concepts which should be mastered. Accordingly, many theorems from these sections can be presented in a simple version or omitted entirely; due to the lack of time, such abridgement is unavoidable. It is up to the instructor to determine how to prevent the lectures from becoming a tedious listing of definitions. We hope that the remaining sections of the course will be of some help in this task. A number of improvements set this paperback edition of our book apart from the first one (Gordon and Breach Science Publishers, 1989). First of all, the terminology was slightly changed in order to be closer to the traditions of western universities. Secondly, the material of some sections was rewritten: for example, the more elaborate section 15 of Chapter 2. While discussing problems of linear programming in section 2 of Chapter 3 the emphasis was changed a little; in particular, we introduced a new example illustrating an application of the theory to microeconomics. Plenty of small corrections were also made to improve the perception of the main theme. We would like to express heartfelt gratitude to Gordon and Breach Science Publishers for taking the initiative that led to the paperback edition of this book. What is more important is that Gordon and Breach prepared LINEAR ALGEBRA AND GEOMETRY ix the publication of Exercises in Algebra by Kostrikin, which is an important addition to Linear Algebra and Geometry, and to Introduction to Algebra. These three books constitute a single algebraic complex, and provide more than enough background for an undergraduate course. A.I. Kostrikin Yu.I. Manin 1996 Bibliography 1 Kostrikin, A. I., (1982) Introduction to Algebra, Springer-Verlag, New York-Berlin 2 Lang, S., (1971) Algebra, Addison-Wesley, Reading. MA 3 Gel'fand, I. M., (1961) Lectures on Linear Algebra, Interscience Publishers, Inc., New York 4Halmos, P. R., (1958) Finite-Dimensional Vector Spaces.
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