Linear Algebra and Geometry

322 Pages·1989·5.48 MB·English

by  Alexei I. Kostrikin

and Geometry
I
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
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Originally published in Russian in 1981 as Jlrraeiiean 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, Linear 2. Geometry
1. Title II. 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
6 Quotient 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
6 Unitary 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
4 Convex 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
9 Desargues' 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
9 Tensor 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
4 Halmos, P. R., (1958) Finite-Dimensional Vector Spaces. 1). Van Nostrand
Company, Inc., New York
5 Artin, E., (1957) Geometric Algebra, Interscience Publishers, Inc., New York
6 Glazman, I. M. and Ljubich, Ju. I., (1974) Finite-Dimensional Linear Analy-
sis: A Systemic Presentation in Problems Form, MIT Press, Cambridge, MA
7 Mansfield, Ed., (1990) Managerial Economics, W. W. Norton & Company,
New York-London
8 Huppert, B., (1990) Angewandte Lineare Algebra, Walter de Gruyter,
Berlin-New York