Preface to the Instructor ix
Preface to the Student xiii
Acknowledgments xv
Chapter 1
Vector Spaces 1
Complex Numbers 2
Definition of Vector Space 4
Properties of Vector Spaces 11
Subspaces 13
Sums and Direct Sums 14
Exercises 19
Chapter 2
Finite-Dimensional Vector Spaces 21
Span and Linear Independence 22
Bases 27
Dimension 31
Exercises 35
Chapter 3
Linear Maps 37
Definitions and Examples 38
Null Spaces and Ranges 41
The Matrix of a Linear Map 48
Invertibility 53
Exercises 59
v
vi Contents
Chapter 4
Polynomials 63
Degree 64
Complex Coefficients 67
Real Coefficients 69
Exercises 73
Chapter 5
Eigenvalues and Eigenvectors 75
Invariant Subspaces 76
Polynomials Applied to Operators 80
Upper-Triangular Matrices 81
Diagonal Matrices 87
Invariant Subspaces on Real Vector Spaces 91
Exercises 94
Chapter 6
Inner-Product Spaces 97
Inner Products 98
Norms 102
Orthonormal Bases 106
Orthogonal Projections and Minimization Problems 111
Linear Functionals and Adjoints 117
Exercises 122
Chapter 7
Operators on Inner-Product Spaces 127
Self-Adjoint and Normal Operators 128
The Spectral Theorem 132
Normal Operators on Real Inner-Product Spaces 138
Positive Operators 144
Isometries 147
Polar and Singular-Value Decompositions 152
Exercises 158
Chapter 8
Operators on Complex Vector Spaces 163
Generalized Eigenvectors 164
The Characteristic Polynomial 168
Decomposition of an Operator 173
Contents vii
Square Roots 177
The Minimal Polynomial 179
Jordan Form 183
Exercises 188
Chapter 9
Operators on Real Vector Spaces 193
Eigenvalues of Square Matrices 194
Block Upper-Triangular Matrices 195
The Characteristic Polynomial 198
Exercises 210
Chapter 10
Trace and Determinant 213
Change of Basis 214
Trace 216
Determinant of an Operator 222
Determinant of a Matrix 225
Volume 236
Exercises 244
Symbol Index 247
Index 249
Preface to the Instructor
You are probably about to teach a course that will give students
their second exposure to linear algebra. During their first brush with
the subject, your students probably worked with Euclidean spaces and
matrices. In contrast, this course will emphasize abstract vector spaces
and linear maps.
The audacious title of this book deserves an explanation. Almost
all linear algebra books use determinants to prove that every linear op-
erator on a finite-dimensional complex vector space has an eigenvalue.
Determinants are difficult, nonintuitive, and often defined without mo-
tivation. To prove the theorem about existence of eigenvalues on com-
plex vector spaces, most books must define determinants, prove that a
linear map is not invertible if and only if its determinant equals 0, and
then define the characteristic polynomial. This tortuous (torturous?)
path gives students little feeling for why eigenvalues must exist.
In contrast, the simple determinant-free proofs presented here of-
fer more insight. Once determinants have been banished to the end
of the book, a new route opens to the main goal of linear algebra—
understanding the structure of linear operators.
This book starts at the beginning of the subject, with no prerequi-
sites other than the usual demand for suitable mathematical maturity.
Even if your students have already seen some of the material in the
first few chapters, they may be unaccustomed to working exercises of
the type presented here, most of which require an understanding of
proofs.
• Vector spaces are defined in Chapter 1, and their basic properties
are developed.
• Linear independence, span, basis, and dimension are defined in
Chapter 2, which presents the basic theory of finite-dimensional
vector spaces.
ix
x Preface to the Instructor
• Linear maps are introduced in Chapter 3. The key result here
is that for a linear map T, the dimension of the null space of T
plus the dimension of the range of T equals the dimension of the
domain of T.
• The part of the theory of polynomials that will be needed to un-
derstand linear operators is presented in Chapter 4. If you take
class time going through the proofs in this chapter (which con-
tains no linear algebra), then you probably will not have time to
cover some important aspects of linear algebra. Your students
will already be familiar with the theorems about polynomials in
this chapter, so you can ask them to read the statements of the
results but not the proofs. The curious students will read some
of the proofs anyway, which is why they are included in the text.
• The idea of studying a linear operator by restricting it to small
subspaces leads in Chapter 5 to eigenvectors. The highlight of the
chapter is a simple proof that on complex vector spaces, eigenval-
ues always exist. This result is then used to show that each linear
operator on a complex vector space has an upper-triangular ma-
trix with respect to some basis. Similar techniques are used to
show that every linear operator on a real vector space has an in-
variant subspace of dimension 1 or 2. This result is used to prove
that every linear operator on an odd-dimensional real vector space
has an eigenvalue. All this is done without defining determinants
or characteristic polynomials!
• Inner-product spaces are defined in Chapter 6, and their basic
properties are developed along with standard tools such as ortho-
normal bases, the Gram-Schmidt procedure, and adjoints. This
chapter also shows how orthogonal projections can be used to
solve certain minimization problems.
• The spectral theorem, which characterizes the linear operators for
which there exists an orthonormal basis consisting of eigenvec-
tors, is the highlight of Chapter 7. The work in earlier chapters
pays off here with especially simple proofs. This chapter also
deals with positive operators, linear isometries, the polar decom-
position, and the singular-value decomposition.
Preface to the Instructor xi
• The minimal polynomial, characteristic polynomial, and general-
ized eigenvectors are introduced in Chapter 8. The main achieve-
ment of this chapter is the description of a linear operator on
a complex vector space in terms of its generalized eigenvectors.
This description enables one to prove almost all the results usu-
ally proved using Jordan form. For example, these tools are used
to prove that every invertible linear operator on a complex vector
space has a square root. The chapter concludes with a proof that
every linear operator on a complex vector space can be put into
Jordan form.
• Linear operators on real vector spaces occupy center stage in
Chapter 9. Here two-dimensional invariant subspaces make up
for the possible lack of eigenvalues, leading to results analogous
to those obtained on complex vector spaces.
• The trace and determinant are defined in Chapter 10 in terms
of the characteristic polynomial (defined earlier without determi-
nants). On complex vector spaces, these definitions can be re-
stated: the trace is the sum of the eigenvalues and the determi-
nant is the product of the eigenvalues (both counting multiplic-
ity). These easy-to-remember definitions would not be possible
with the traditional approach to eigenvalues because that method
uses determinants to prove that eigenvalues exist. The standard
theorems about determinants now become much clearer. The po-
lar decomposition and the characterization of self-adjoint opera-
tors are used to derive the change of variables formula for multi-
variable integrals in a fashion that makes the appearance of the
determinant there seem natural.
This book usually develops linear algebra simultaneously for real
and complex vector spaces by letting F denote either the real or the
complex numbers. Abstract fields could be used instead, but to do so
would introduce extra abstraction without leading to any new linear al-
gebra. Another reason for restricting attention to the real and complex
numbers is that polynomials can then be thought of as genuine func-
tions instead of the more formal objects needed for polynomials with
coefficients in finite fields. Finally, even if the beginning part of the the-
ory were developed with arbitrary fields, inner-product spaces would
push consideration back to just real and complex vector spaces.
xii Preface to the Instructor
Even in a book as short as this one, you cannot expect to cover every-
thing. Going through the first eight chapters is an ambitious goal for a
one-semester course. If you must reach Chapter 10, then I suggest cov-
ering Chapters 1, 2, and 4 quickly (students may have seen this material
in earlier courses) and skipping Chapter 9 (in which case you should
discuss trace and determinants only on complex vector spaces).
A goal more important than teaching any particular set of theorems
is to develop in students the ability to understand and manipulate the
objects of linear algebra. Mathematics can be learned only by doing;
fortunately, linear algebra has many good homework problems. When
teaching this course, I usually assign two or three of the exercises each
class, due the next class. Going over the homework might take up a
third or even half of a typical class.
A solutions manual for all the exercises is available (without charge)
only to instructors who are using this book as a textbook. To obtain
the solutions manual, instructors should send an e-mail request to me
(or contact Springer if I am no longer around).
Please check my web site for a list of errata (which I hope will be
empty or almost empty) and other information about this book.
I would greatly appreciate hearing about any errors in this book,
even minor ones. I welcome your suggestions for improvements, even
tiny ones. Please feel free to contact me.
Have fun!
Sheldon Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132, USA
e-mail: axler@math.sfsu.edu
www home page: http://math.sfsu.edu/axler
Preface to the Student
You are probably about to begin your second exposure to linear al-
gebra. Unlike your first brush with the subject, which probably empha-
sized Euclidean spaces and matrices, we will focus on abstract vector
spaces and linear maps. These terms will be defined later, so don’t
worry if you don’t know what they mean. This book starts from the be-
ginning of the subject, assuming no knowledge of linear algebra. The
key point is that you are about to immerse yourself in serious math-
ematics, with an emphasis on your attaining a deep understanding of
the definitions, theorems, and proofs.
You cannot expect to read mathematics the way you read a novel. If
you zip through a page in less than an hour, you are probably going too
fast. When you encounter the phrase “as you should verify”, you should
indeed do the verification, which will usually require some writing on
your part. When steps are left out, you need to supply the missing
pieces. You should ponder and internalize each definition. For each
theorem, you should seek examples to show why each hypothesis is
necessary.
Please check my web site for a list of errata (which I hope will be
empty or almost empty) and other information about this book.
I would greatly appreciate hearing about any errors in this book,
even minor ones. I welcome your suggestions for improvements, even
tiny ones.
Have fun!
Sheldon Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132, USA
e-mail: axler@math.sfsu.edu
www home page: http://math.sfsu.edu/axler
xiii
Acknowledgments
I owe a huge intellectual debt to the many mathematicians who cre-
ated linear algebra during the last two centuries. In writing this book I
tried to think about the best way to present linear algebra and to prove
its theorems, without regard to the standard methods and proofs used
in most textbooks. Thus I did not consult other books while writing
this one, though the memory of many books I had studied in the past
surely influenced me. Most of the results in this book belong to the
common heritage of mathematics. A special case of a theorem may
first have been proved in antiquity (which for linear algebra means the
nineteenth century), then slowly sharpened and improved over decades
by many mathematicians. Bestowing proper credit on all the contrib-
utors would be a difficult task that I have not undertaken. In no case
should the reader assume that any theorem presented here represents
my original contribution.
Many people helped make this a better book. For useful sugges-
tions and corrections, I am grateful to William Arveson (for suggesting
the proof of 5.13), Marilyn Brouwer, William Brown, Robert Burckel,
Paul Cohn, James Dudziak, David Feldman (for suggesting the proof of
8.40), Pamela Gorkin, Aram Harrow, Pan Fong Ho, Dan Kalman, Robert
Kantrowitz, Ramana Kappagantu, Mizan Khan, Mikael Lindstr
¨
om, Ja-
cob Plotkin, Elena Poletaeva, Mihaela Poplicher, Richard Potter, Wade
Ramey, Marian Robbins, Jonathan Rosenberg, Joan Stamm, Thomas
Starbird, Jay Valanju, and Thomas von Foerster.
Finally, I thank Springer for providing me with help when I needed
it and for allowing me the freedom to make the final decisions about
the content and appearance of this book.
xv
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