# A Course in Linear Algebra With Applications 2nd Edition by Derek J. S. Robinson

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Size 11.1 MiB ## PREFACE TO THE SECOND EDITION

The principal change from the first edition is the addition of
a new chapter on linear programming. While linear programming
is one of the most widely used and successful applications
of linear algebra, it rarely appears in a text such as this. In
the new Chapter Ten the theoretical basis of the simplex algorithm
is carefully explained and its geometrical interpretation
is stressed.
Some further applications of linear algebra have been
added, for example the use of Jordan normal form to solve
systems of linear differential equations and a discussion of extremal
On the theoretical side, the concepts of coset and quotient
space are thoroughly explained in Chapter 5. Cosets have
useful interpretations as solutions sets of systems of linear
equations. In addition the Isomorphisms Theorems for vector
spaces are developed in Chapter Six: these shed light on the
relationship between subspaces and quotient spaces.
The opportunity has also been taken to add further exercises,
revise the exposition in several places and correct a few
errors. Hopefully these improvements will increase the usefulness
of the book to anyone who needs to have a thorough
knowledge of linear algebra and its applications.
I am grateful to Ms. Tan Rok Ting of World Scientific
for assistance with the production of this new edition and for
patience in the face of missed deadlines. I thank my family
for their support during the preparation of the manuscript.

## PREFACE TO THE FIRST EDITION

A rough and ready definition of linear algebra might be: that
part of algebra which is concerned with quantities of the first
degree. Thus, at the very simplest level, it involves the solution
of systems of linear equations, and in a real sense this
elementary problem underlies the whole subject. Of all the
branches of algebra, linear algebra is the one which has found
the widest range of applications. Indeed there are few areas
of the mathematical, physical and social sciences which have
not benefitted from its power and precision. For anyone working
in these fields a thorough knowledge of linear algebra has
become an indispensable tool. A recent feature is the greater
mathematical sophistication of users of the subject, due in
part to the increasing use of algebra in the information sciences.
At any rate it is no longer enough simply to be able to
perform Gaussian elimination and deal with real vector spaces
of dimensions two and three.
The aim of this book is to give a comprehensive introduction
to the core areas of linear algebra, while at the same
time providing a selection of applications. We have taken the
point of view that it is better to consider a few quality applications
in depth, rather than attempt the almost impossible task
of covering all conceivable applications that potential readers
might have in mind.
The reader is not assumed to have any previous knowledge
of linear algebra – though in practice many will – but
is expected to have at least the mathematical maturity of a
student who has completed the calculus sequence. In North
America such a student will probably be in the second or third
year of study.
The book begins with a thorough discussion of matrix
operations. It is perhaps unfashionable to precede systems
of linear equations by matrices, but I feel that the central
position of matrices in the entire theory makes this a logical
and reasonable course. However the motivation for the introduction
of matrices, by means of linear equations, is still
provided informally. The second chapter forms a basis for
the whole subject with a full account of the theory of linear
equations. This is followed by a chapter on determinants, a
topic that has been unfairly neglected recently. In practice it
is hard to give a satisfactory definition of the general n x n
determinant without using permutations, so a brief account
of these is given.
Chapters Five and Six introduce the student to vector
spaces. The concept of an abstract vector space is probably
the most challenging one in the entire subject for the nonmathematician,
but it is a concept which is well worth the
effort of mastering. Our approach proceeds in gentle stages,
through a series of examples that exhibit the essential features
of a vector space; only then are the details of the definition
written down. However I feel that nothing is gained
by ducking the issue and omitting the definition entirely, as is
sometimes done.
Linear tranformations are the subject of Chapter Six.
After a brief introduction to functional notation, and numerous
examples of linear transformations, a thorough account
of the relation between linear transformations and matrices is
given. In addition both kernel and image are introduced and
are related to the null and column spaces of a matrix.
Orthogonality, perhaps the heart of the subject, receives
an extended treatment in Chapter Seven. After a gentle introduction
by way of scalar products in three dimensions —
which will be familiar to the student from calculus — inner
product spaces are denned and the Gram-Schmidt procedure
is described. The chapter concludes with a detailed account
of The Method of Least Squares, including the problem of
finding optimal solutions, which texts at this level often fail
to cover.
Chapter Eight introduces the reader to the theory of
eigenvectors and eigenvalues, still one of the most powerful
tools in linear algebra. Included is a detailed account of applications
to systems of linear differential equations and linear
recurrences, and also to Markov processes. Here we have not
shied away from the more difficult case where the eigenvalues
of the coefficient matrix are not all different.
The final chapter contains a selection of more advanced
topics in linear algebra, including the crucial Spectral Theorem
on the diagonalizability of real symmetric matrices. The
usual applications of this result to quadratic forms, conies
and quadrics, and maxima and minima of functions of several
variables follow.
Also included in Chapter Nine are treatments of bilinear
forms and Jordan Normal Form, topics that are often not considered
in texts at this level, but which should be more widely
known. In particular, canonical forms for both symmetric and
skew-symmetric bilinear forms are obtained. Finally, Jordan
Normal Form is presented by an accessible approach that requires
only an elementary knowledge of vector spaces.
Chapters One to Eight, together with Sections 9.1 and
9.2, correspond approximately to a one semester course taught
by the author over a period of many years. As time allows,
other topics from Chapter Nine may be included. In practice
some of the contents of Chapters One and Two will already be
familiar to many readers and can be treated as review. Full
proofs are almost always included: no doubt some instructors
may not wish to cover all of them, but it is stressed that for
maximum understanding of the material as many proofs as
possible should be read. A good supply of problems appears
at the end of each section. As always in mathematics, it is an
indispensible part of learning the subject to attempt as many
problems as possible.
This book was originally begun at the suggestion of
Harriet McQuarrie. I thank Ms. Ho Hwei Moon of World
Scientific Publishing Company for her advice and for help with
editorial work. I am grateful to my family for their patience,
and to my wife Judith for her encouragement, and for assistance

## CONTENTS

Preface to the Second Edition vii
Preface to the First Edition ix
Chapter One Matrix Algebra
1.1 Matrices 1
1.2 Operations with Matrices 6
1.3 Matrices over Rings and Fields 24
Chapter Two Systems of Linear Equations
2.1 Gaussian Elimination 30
2.2 Elementary Row Operations 41
2.3 Elementary Matrices 47
Chapter Three Determinants
3.1 Permutations and the Definition of a
Determinant 57
3.2 Basic Properties of Determinants 70
3.3 Determinants and Inverses of Matrices 78
Chapter Four Introduction to Vector Spaces
4.1 Examples of Vector Spaces 87
4.2 Vector Spaces and Subspaces 95
4.3 Linear Independence in Vector Spaces 104
Chapter Five Basis and Dimension
5.1 The Existence of a Basis 112
5.2 The Row and Column Spaces of a Matrix 126
5.3 Operations with Subspaces 133
Chapter Six Linear Transformations
6.1 Functions Defined on Sets 152
6.2 Linear Transformations and Matrices 158
6.3 Kernel, Image and Isomorphism 178
Chapter Seven Orthogonality in Vector Spaces
7.1 Scalar Products in Euclidean Space 193
7.2 Inner Product Spaces 209
7.3 Orthonormal Sets and the Gram-Schmidt
Process 226
7.4 The Method of Least Squares 241
Chapter Eight Eigenvectors and Eigenvalues
8.1 Basic Theory of Eigenvectors and Eigenvalues 257
8.2 Applications to Systems of Linear Recurrences 276
8.3 Applications to Systems of Linear Differential
Equations 288
9.1 Eigenvalues and Eigenvectors of Symmetric and
Hermitian Matrices 303
9.3 Bilinear Forms 332
9.4 Minimum Polynomials and Jordan Normal
Form 347
Chapter Ten Linear Programming
10.1 Introduction to Linear Programming 370
10.2 The Geometry of Linear Programming 380
10.3 Basic Solutions and Extreme Points 391
10.4 The Simplex Algorithm 399
Appendix Mathematical Induction 415