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首页线性代数新解:结构与证明中心
"《线性代数做得正确》(Linear Algebra Done Right)是一本面向数学专业学生和研究生的第二课程教材,由Sheldon Axler撰写,属于Undergraduate Texts in Mathematics系列。该书以其独特的教学方法脱颖而出,不将行列式作为核心内容,而是将重点放在理解向量空间上的线性运算结构上。作者注重概念的动机和证明的简化,例如在未定义行列式的情况下,书中提供了一个清晰的证明,即任何有限维复数向量空间上的线性算子都至少有一个特征值。 书本从基本概念开始,如向量空间、线性独立、span和维度,为读者建立起扎实的基础。前半部分讨论内积空间,随后引入了有限维谱定理。每一章都包含丰富多样的习题,帮助学生深入理解和操作线性代数的对象。第三版新增了关于对角矩阵、线性泛函和共轭转置、以及谱定理的章节,一些部分如自伴和正规算子的论述进行了全面重写,并对全文进行了数百处的细微改进。 Undergraduate Texts in Mathematics系列的目标是为北美大学的三年级和四年级数学学生提供新颖的教学资源,旨在提供学生和教师全新的视角,以便他们能更好地掌握线性代数的基本原理和应用技巧。通过《线性代数做得正确》这本书,学生们不仅能学习到严谨的理论,还能培养抽象思维和问题解决能力,为今后的专业发展打下坚实基础。" 《线性代数做得正确》是一部注重概念理解和证明简洁性的教科书,它的独特之处在于它将行列式置于次要位置,而强调了线性代数的核心概念和实际应用,使得学习者能够从更深入的角度理解和掌握线性代数这一重要的数学分支。
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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
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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
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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
R
n
and
C
n
, 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
R
n
and C
n
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
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2 CHAPTER 1 Vector Spaces
1.A R
n
and C
n
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 a C bi.
The set of all complex numbers is denoted by C:
C Dfa C bi W a; b 2 Rg:
Addition and multiplication on C are defined by
.a C bi/ C .c C di/ D .a C c/ C .b C d/i;
.a C bi/.c C di/ D .ac bd/ C .ad Cbc/iI
here a; b; c; d 2 R.
If
a 2 R
, we identify
a C 0i
with the real number
a
. Thus we can think
of
R
as a subset of
C
. We also usually write
0 Cbi
as just
bi
, and we usually
write 0 C 1i as just i.
The symbol
i
was first used to de-
note
p
1
by Swiss mathematician
Leonhard Euler in 1777.
Using multiplication as defined
above, you should verify that
i
2
D1
.
Do not memorize the formula for the
product of two complex numbers; you
can always rederive it by recalling that
i
2
D1
and then using the usual rules
of arithmetic (as given by 1.3).
1.2 Example Evaluate .2 C 3i /.4 C 5i/.
Solution
.2 C 3i/.4 C 5i/ D 2 4 C 2 .5i/ C .3i / 4 C .3i/.5i /
D 8 C 10i C 12i 15
D7 C 22i
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SECTION 1.A R
n
and C
n
3
1.3 Properties of complex arithmetic
commutativity
˛ C ˇ D ˇ C ˛ and ˛ˇ D ˇ˛ for all ˛; ˇ 2 C;
associativity
.˛ Cˇ/C D ˛ C.ˇ C/
and
.˛ˇ/ D ˛.ˇ/
for all
˛; ˇ; 2 C
;
identities
C 0 D and 1 D for all 2 C;
additive inverse
for every
˛ 2 C
, there exists a unique
ˇ 2 C
such that
˛ C ˇ D 0
;
multiplicative inverse
for every
˛ 2 C
with
˛ ¤ 0
, there exists a unique
ˇ 2 C
such that
˛ˇ D 1;
distributive property
.˛ C ˇ/ D ˛ C ˇ for all ;˛;ˇ 2 C.
The properties above are proved using the familiar properties of real
numbers and the definitions of complex addition and multiplication. The
next example shows how commutativity of complex multiplication is proved.
Proofs of the other properties above are left as exercises.
1.4 Example Show that ˛ˇ D ˇ˛ for all ˛; ˇ; 2 C.
Solution
Suppose
˛ D a C bi
and
ˇ D c C di
, where
a; b; c; d 2 R
. Then
the definition of multiplication of complex numbers shows that
˛ˇ D .a C bi/.c C di/
D .ac bd/ C .ad C bc/i
and
ˇ˛ D .c C d i/.a C bi/
D .ca db/ C .cb C da/i:
The equations above and the commutativity of multiplication and addition of
real numbers show that ˛ˇ D ˇ˛.
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