Linear Algebra: Examples and Applications

386 Pages·2005·13.085 MB·English

by  Alain M. Robert

LINEAR
ALGEBRA
Examples and Applications
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LINEAR
ALGEBRA
Examples and Applications
Alain M. Robert
Universite de NeuchBtel, Switzerland
\;
Sworid Scientific
NEW JERSEY LONDON*SINGAPORE*SHANGHAl *HONG KONG*TAIPEI*BANGALORE
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LINEAR ALGEBRA
Examples and Applications
Copyright 0 2005 by World Scientific Publishing Co. Re. Ltd.
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ISBN 981-256-432-2
ISBN 981-256-499-3 (pbk)
Printed in Singapore by B & JO Enterprise
To the memory of my parents,
who understood that
I would never make a good farmer
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Foreword
Mathematics is a living creation, and linear algebra has undergone a real meta-
morphosis during the twentieth century, partly due to the birth and development
of computers. It is so active that entire periodical magazines are now devoted
to it, and one single book can only reflect part of its vitality. Here is an at-
tempt to face this challenge in a concise-although rigorous-manner. Linear
algebra is a general and powerful language. This book is based on examples and
applications, justifying the elaboration of such an abstract language.
In the first part, vector spaces are approached through carefully chosen
linear systems, and linear maps are introduced through matrix multiplication.
The four initial chapters constitute the skeleton of the linear category. The
importance and ubiquity of this structure is emphasized by the applications of
the rank theory (Chapter 5), and in the geometric approach to eigenvectors
(Chapter 6). Since even and odd functions appear as the eigenspaces of the
symmetry operator, we do not assume a priori finite dimensionality, and bases
are discussed and examples are given in the general context.
The second part is devoted to the study of metric relations (angles, orthog-
onality) in real vector spaces. Several geometric properties can easily be derived
from an inner product. The best approximation theorem, with its application
to the mean squares method is certainly the most used in practice. Here bi-
linearity appears on the scene, and this fascinating property culminates in the
abstract form of duality (Chapter 9).
Finally, the third part is rooted in volume computations, revealing the
phenomenon of multi-linearity. Hence, determinants come last (Chapter lo),
and constitute the golden adornment of the theory. They play an essential part
in the algebraic properties of eigenvalues. The main result proved in this book is
the spectral theorem (for real symmetric matrices in Chapter 8, and for normal
operators in Chapter 12). Its geometrical meaning is emphasized with the polar
decomposition for linear maps between finite-dimensional real vector spaces.
A few appendices contain independent complements. Of special importance
is the appendix on finite probability spaces, where the notion of independence
for random variables is compared with that of linear independence.
As is probably apparent, this book is written for curious and motivated stu-
dents in physics, chemistry, computer science, engineering,. . . and not solely for
vii
...
FOREWORD
Vlll
mathematicians. I believe that our duty is to form scientists capable of under-
standing each other’s problems. Having in the same class students interested
in various disciplines provides an opportunity to show them the relevance of
mathematics through linear algebra, by selecting examples that might catch
their interest. It should not be wasted on teaching them to perform mechanical
manipulations based on a set of axioms, a task better suited to a computer! This
is why I have tried to minimize the axiomatic aspect, leaving out the discussion
of general fields, assuming implicitly that the scalars are real (or complex) num-
bers. But I have chosen general proofs of the main theorems (and in particular
for the rank theorem), relegating the use of inner products and orthogonality
(specific to real numbers) to the second part of the book, as already mentioned.
These students are supposed to have a previous acquaintance with basic
calculus and to be familiar with the language of arrows for maps, their compo-
sition, and inversion. Only a brief summary of set theory is included. Another
prerequisite concerns vectors in two and three dimensions, Cartesian and polar
coordinates (elementary trigonometry). Hence this text is directed to students
who follow (or have previously followed) a first calculus course. This is particu-
larly apparent with the examples concerning polynomials and their derivatives,
linear fractional transformations, and rational functions.
Needless to say, exercises, tutorials (or individual support in any form) are
essential to check that the students understand and can apply this theory. Since
books with many routine exercises are easily available, I have limited the number
of such exercises. On the other hand, more difficult problems have been included
(with hints, or even complete solutions).
If I have tried to bring the main facts to the forefront, 1h ave made no effort to
satisfy all the needs of future research mathematicians, or theoretical physicists.
They will have to complete this study by examining vector spaces over any field
(or even modules over principal ideal domains) and tensor products.
I have chosen to avoid the discussion of the normal Jordan form. In my
opinion, its importance is best revealed with a specific application in mind:
Markov chain theory, coupled linear differential systems, Riesz theory for com-
pact operators in Banach spaces, linear algebraic groups (where additive and
multiplicative Jordan decompositions both appear); each provides such an op-
portunity. My purpose was only to convey the basic aspects of this cornerstone
in mathematical education.
Finally, I have to thank 0. Besson, A. Gertsch Hamadene, and A. Junod
who read parts of preliminary versions of this book, detected several mistakes
and made useful suggestions.
April 2005 Alain M. Robert
Contents
Foreword vii
1 Linear Systems: Elimination Method 1
1.1 Examples of Linear Systems . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 A Review Example . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Covering a Sphere with Hexagons and Pentagons . . . . . 2
1.1.3 A Literal Example . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Homogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 A Chemical Reaction . . . . . . . . . . . . . . . . . . . . . 11
1.2.2 Reduced Forms . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 Elimination Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.1 Elementary Row Operations . . . . . . . . . . . . . . . . . 18
1.3.2 Comparison of the Systems (S) and (HS) . . . . . . . . . 21
1.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.4.1 Potentials on a Grid . . . . . . . . . . . . . . . . . . . . . 22
1.4.2 Another Illustration of the Fundamental Principle . . . . 23
+ +
1.4.3 The Euler Theorem f v = e 2 . . . . . . . . . . . . . 25
1.4.4 F’ullerenes, Radiolarians . . . . . . . . . . . . . . . . . . . 25
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2 Vector Spaces 31
2.1 TheLanguage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1.1 Axiomatic Properties . . . . . . . . . . . . . . . . . . . . . 31
2.1.2 An Important Principle . . . . . . . . . . . . . . . . . . . 32
2.1.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.4 Vector Subspaces . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 Finitely Generated Vector Spaces . . . . . . . . . . . . . . . . . . 36
2.2.1 Generators . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.2.2 Linear Independence . . . . . . . . . . . . . . . . . . . . . 39
2.2.3 The Dimension . . . . . . . . . . . . . . . . . . . . . . . . 41
2.3 Infinite-Dimensional Vector Spaces . . . . . . . . . . . . . . . . . 44
2.3.1 The Space of Polynomials . . . . . . . . . . . . . . . . . . 45
2.3.2 Existence of Bases: The Mathematical Credo . . . . . . . 47
2.3.3 Infinite-Dimensional Examples . . . . . . . . . . . . . . . 49
ix
X CONTENTS
2.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.1 Set Theory, Notation . . . . . . . . . . . . . . . . . . . . . 52
2.4.2 Axioms for Fields of Scalars . . . . . . . . . . . . . . . . . 56
2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3 Matrix Multiplication 60
3.1 Row by Column Multiplication . . . . . . . . . . . . . . . . . . . 60
3.1.1 Linear Fractional Transformations . . . . . . . . . . . . . 60
3.1.2 Linear Changes of Variables . . . . . . . . . . . . . . . . . 61
3.1.3 Definition of the Matrix Product . . . . . . . . . . . . . . 62
3.1.4 The Map Produced by Matrix Multiplication . . . . . . . 66
3.2 Row Operations and Matrix Multiplication . . . . . . . . . . . . 67
3.2.1 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . 68
3.2.2 An Inversion Algorithm . . . . . . . . . . . . . . . . . . . 70
3.2.3 LU Factorizations . . . . . . . . . . . . . . . . . . . . . . 72
3.2.4 Simultaneous Resolution of Linear Systems . . . . . . . . 76
3.3 Matrix Multiplication by Blocks . . . . . . . . . . . . . . . . . . 76
3.3.1 Explanation of the Method . . . . . . . . . . . . . . . . . 76
3.3.2 The Field of Complex Numbers . . . . . . . . . . . . . . . 79
3.4 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.1 Afine Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.2 The Field of Quaternions . . . . . . . . . . . . . . . . . . 81
3.4.3 The Strassen Algorithm . . . . . . . . . . . . . . . . . . . 82
3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4 Linear Maps 88
4.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.1.1 Preliminary Considerations . . . . . . . . . . . . . . . . . 88
4.1.2 Definition and First Properties . . . . . . . . . . . . . . . 90
4.1.3 Examples of Linear Maps . . . . . . . . . . . . . . . . . . 91
4.2 General Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2.1 Image and Kernel of a Linear Map . . . . . . . . . . . . . 92
4.2.2 How to Construct Linear Maps . . . . . . . . . . . . . . . 94
4.2.3 Matrix Description of Linear Maps . . . . . . . . . . . . . 95
4.3 The Dimension Theorem for Linear Maps . . . . . . . . . . . . . 98
4.3.1 The Rank-Nullity Theorem . . . . . . . . . . . . . . . . . 98
4.3.2 Row-Rank versus Column-Rank . . . . . . . . . . . . . . 99
4.3.3 Application: Invertible Matrices . . . . . . . . . . . . . . 101
4.4 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.4.2 Models of Finite-Dimensional Vector Spaces . . . . . . . . 1 04
4.4.3 Change of Basis: Components of Vectors . . . . . . . . . 105
4.4.4 Change of Basis: Matrices of Linear Maps . . . . . . . . . 107
4.4.5 The Trace of Square Matrices . . . . . . . . . . . . . . . . 107
4.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.5.1 Inverting Maps Between Sets . . . . . . . . . . . . . . . . 108