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Linear Algebra Done Right

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the Instructor
Major changes from the previous edition:
� This edition contains 561 exercises, including 337 new exercises that were
not in the previous edition. Exercises now appear at the end of each section,
rather than at the end of each chapter.
� Many new examples have been added to illustrate the key ideas of linear
algebra.
� Beautiful new formatting, including the use of color, creates pages with an
unusually pleasant appearance in both print and electronic versions. As a
visual aid, definitions are in beige boxes and theorems are in blue boxes (in
color versions of the book).
� Each theorem now has a descriptive name.
� New topics covered in the book include product spaces, quotient spaces,
and duality.
� Chapter 9 (Operators on Real Vector Spaces) has been completely rewritten
to take advantage of simplifications via complexification. This approach
allows for more streamlined presentations in Chapters 5 and 7 because
those chapters now focus mostly on complex vector spaces.
� Hundreds of improvements have been made throughout the book. For
example, the proof of Jordan Form (Section 8.D) has been simplified.
Please check the website below for additional information about the book. I
may occasionally write new sections on additional topics. These new sections
will be posted on the website. Your suggestions, comments, and corrections
are most welcome.
Best wishes for teaching a successful linear algebra class!
Sheldon Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132, USA
website: linear.axler.net
e-mail: linear@axler.net
Twitter: @AxlerLinear
Preface for the Student
You are probably about to begin your second exposure to linear algebra. Unlike
your first brush with the subject, which probably emphasized Euclidean spaces
and matrices, this encounter will focus on abstract vector spaces and linear
maps. These terms will be defined later, so don’t worry if you do not know
what they mean. This book starts from the beginning of the subject, assuming
no knowledge of linear algebra. The key point is that you are about to
immerse yourself in serious mathematics, with an emphasis on attaining a
deep understanding of the definitions, theorems, and proofs.
You cannot 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. Discussions with other students should help.
As a visual aid, definitions are in beige boxes and theorems are in blue
boxes (in color versions of the book). Each theorem has a descriptive name.
Please check the website below for additional information about the book. I
may occasionally write new sections on additional topics. These new sections
will be posted on the website. Your suggestions, comments, and corrections
are most welcome.
Best wishes for success and enjoyment in learning linear algebra!
Sheldon Axler
Mathematics Department
San Francisco State University
San Francisco, CA 94132, USA
website: linear.axler.net
e-mail: linear@axler.net
Twitter: @AxlerLinear
xv
Acknowledgments
I owe a huge intellectual debt to the many mathematicians who created linear
algebra over the past two centuries. The results in this book belong to the
common heritage of mathematics. A special case of a theorem may first have
been proved in the nineteenth century, then slowly sharpened and improved by
many mathematicians. Bestowing proper credit on all the contributors 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.
However, in writing this book I tried to think about the best way to present lin-
ear algebra and to prove its theorems, without regard to the standard methods
and proofs used in most textbooks.
Many people helped make this a better book. The two previous editions
of this book were used as a textbook at about 300 universities and colleges. I
received thousands of suggestions and comments from faculty and students
who used the second edition. I looked carefully at all those suggestions as I
was working on this edition. At first, I tried keeping track of whose suggestions
I used so that those people could be thanked here. But as changes were made
and then replaced with better suggestions, and as the list grew longer, keeping
track of the sources of each suggestion became too complicated. And lists are
boring to read anyway. Thus in lieu of a long list of people who contributed
good ideas, I will just say how truly grateful I am to everyone who sent me
suggestions and comments. Many many thanks!
Special thanks to Ken Ribet and his giant (220 students) linear algebra
class at Berkeley that class-tested a preliminary version of this third edition
and that sent me more suggestions and corrections than any other group.
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. Special thanks to Elizabeth Loew for her wonderful
work as editor and David Kramer for unusually skillful copyediting.
Sheldon Axler
xvii
CHAPTER
1
René Descartes explaining his
work to Queen Christina of
Sweden. Vector spaces are a
generalization of the
description of a plane using
two coordinates, as published
by Descartes in 1637.
Vector Spaces
Linear algebra is the study of linear maps on finite-dimensional vector spaces.
Eventually we will learn what all these terms mean. In this chapter we will
define vector spaces and discuss their elementary properties.
In linear algebra, better theorems and more insight emerge if complex
numbers are investigated along with real numbers. Thus we will begin by
introducing the complex numbers and their basic properties.
We will generalize the examples of a plane and ordinary space to Rn
and Cn, which we then will generalize to the notion of a vector space. The
elementary properties of a vector space will already seem familiar to you.
Then our next topic will be subspaces, which play a role for vector spaces
analogous to the role played by subsets for sets. Finally, we will look at sums
of subspaces (analogous to unions of subsets) and direct sums of subspaces
(analogous to unions of disjoint sets).
LEARNING OBJECTIVES FOR THIS CHAPTER
basic properties of the complex numbers
Rn and Cn
vector spaces
subspaces
sums and direct sums of subspaces
© Springer International Publishing 2015
S. Axler, Linear Algebra Done Right, Undergraduate Texts in Mathematics,
DOI 10.1007/978-3-319-11080-6__1
1
2 CHAPTER 1 Vector Spaces
1.A Rn and Cn
Complex Numbers
You should already be familiar with basic properties of the set R of real
numbers. Complex numbers were invented so that we can take square roots of
negative numbers. The idea is to assume we have a square root of �1, denoted
i , that obeys the usual rules of arithmetic. Here are the formal definitions:
1.1 Definition complex numbers
� A complex number is an ordered pair .a; b/, where a; b 2 R, but
we will write this as aC bi .
� The set of all complex numbers is denoted by C:
C D faC bi W a; b 2 Rg:
� Addition and multiplication on C are defined by
.aC bi/C .c C di/ D .aC c/C .b C d/i;
.aC bi/.c C di/ D .ac � bd/C .ad C bc/i I
here a; b; c; d 2 R.
If a 2 R, we identify aC 0i with the real number a. Thus we can think
of R as a subset of C. We also usually write 0C bi as just bi , and we usually
write 0C 1i as just i .
The symbol i was