# 线性代数（来源于英文维基教科书）

Linear Algebra 线性代数
An Introduction to Mathematical Discourse 数学话语导论

The book was designed specifically for students who had not previously been exposed to mathematics as mathematicians view it. That is, as a subject whose goal is to rigorously prove theorems starting from clear consistent definitions. This book attempts to build students up from a background where mathematics is simply a tool that provides useful calculations to the point where the students have a grasp of the clear and precise nature of mathematics. A more detailed discussion of the prerequisites and goal of this book is given in the introduction.

### 符号说明

 $\mathbb {R}$ , $\mathbb {R} ^{+}$ , $\mathbb {R} ^{n}$ real numbers, reals greater than $0$ , ordered $n$ -tuples of reals $\mathbb {N}$ natural numbers: $\{0,1,2,\ldots \}$ $\mathbb {C}$ complex numbers $\{\ldots \,{\big |}\,\ldots \}$ set of . . . such that . . . $(a\,..\,b)$ , $[a\,..\,b]$ interval (open or closed) of reals between $a$ and $b$ $\langle \ldots \rangle$ sequence; like a set but order matters $V,W,U$ vector spaces ${\vec {v}},{\vec {w}}$ vectors ${\vec {0}}$ , ${\vec {0}}_{V}$ zero vector, zero vector of $V$ $B,D$ bases ${\mathcal {E}}_{n}=\langle {\vec {e}}_{1},\,\ldots ,\,{\vec {e}}_{n}\rangle$ standard basis for $\mathbb {R} ^{n}$ ${\vec {\beta }},{\vec {\delta }}$ basis vectors ${\rm {Rep}}_{B}({\vec {v}})$ matrix representing the vector ${\mathcal {P}}_{n}$ set of $n$ -th degree polynomials ${\mathcal {M}}_{n\!\times \!m}$ set of $n\!\times \!m$ matrices $[S]$ span of the set $S$ $M\oplus N$ direct sum of subspaces $V\cong W$ isomorphic spaces $h,g$ homomorphisms, linear maps $H,G$ matrices $t,s$ transformations; maps from a space to itself $T,S$ square matrices ${\rm {Rep}}_{B,D}(h)$ matrix representing the map $h$ $h_{i,j}$ matrix entry from row $i$ , column $j$ $\left|T\right|$ determinant of the matrix $T$ 矩阵$T$ 的行列式 ${\mathcal {R}}(h),{\mathcal {N}}(h)$ rangespace and nullspace of the map $h$ ${\mathcal {R}}_{\infty }(h),{\mathcal {N}}_{\infty }(h)$ generalized rangespace and nullspace

#### Lower case Greek alphabet 小写希腊字母

${\begin{array}{ll|ll|ll}{\text{name}}&{\text{character}}&{\text{name}}&{\text{character}}&{\text{name}}&{\text{character}}\\\hline {\text{alpha}}&\alpha &{\text{iota}}&\iota &{\text{rho}}&\rho \\{\text{beta}}&\beta &{\text{kappa}}&\kappa &{\text{sigma}}&\sigma \\{\text{gamma}}&\gamma &{\text{lambda}}&\lambda &{\text{tau}}&\tau \\{\text{delta}}&\delta &{\text{mu}}&\mu &{\text{upsilon}}&\upsilon \\{\text{epsilon}}&\epsilon &{\text{nu}}&\nu &{\text{phi}}&\phi \\{\text{zeta}}&\zeta &{\text{xi}}&\xi &{\text{chi}}&\chi \\{\text{eta}}&\eta &{\text{omicron}}&o&{\text{psi}}&\psi \\{\text{theta}}&\theta &{\text{pi}}&\pi &{\text{omega}}&\omega \end{array}}$ About the Cover. This is Cramer's Rule for the system $x_{1}+2x_{2}=6$ , $3x_{1}+x_{2}=8$ . The size of the first box is the determinant shown (the absolute value of the size is the area). The size of the second box is $x_{1}$ times that, and equals the size of the final box. Hence, $x_{1}$ is the final determinant divided by the first determinant.

### 介绍

This book helps students to master the material of a standard undergraduate linear algebra course.

The material is standard in that the topics covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. The audience is also standard: sophomores or juniors, usually with a background of at least one semester of calculus and perhaps with as much as three semesters.

The help that it gives to students comes from taking a developmental approach—this book's presentation emphasizes motivation and naturalness, driven home by a wide variety of examples and extensive, careful, exercises. The developmental approach is what sets this book apart, so some expansion of the term is appropriate here.

Courses in the beginning of most mathematics programs reward students less for understanding the theory and more for correctly applying formulas and algorithms. Later courses ask for mathematical maturity: the ability to follow different types of arguments, a familiarity with the themes that underlay many mathematical investigations like elementary set and function facts, and a capacity for some independent reading and thinking. Linear algebra is an ideal spot to work on the transition between the two kinds of courses. It comes early in a program so that progress made here pays off later, but also comes late enough that students are often majors and minors. The material is coherent, accessible, and elegant. There are a variety of argument styles—proofs by contradiction, if and only if statements, and proofs by induction, for instance—and examples are plentiful.

So, the aim of this book's exposition is to help students develop from being successful at their present level, in classes where a majority of the members are interested mainly in applications in science or engineering, to being successful at the next level, that of serious students of the subject of mathematics itself.

Helping students make this transition means taking the mathematics seriously, so all of the results in this book are proved. On the other hand, we cannot assume that students have already arrived, and so in contrast with more abstract texts, we give many examples and they are often quite detailed.

In the past, linear algebra texts commonly made this transition abruptly. They began with extensive computations of linear systems, matrix multiplications, and determinants. When the concepts—vector spaces and linear maps—finally appeared, and definitions and proofs started, often the change brought students to a stop. In this book, while we start with a computational topic, linear reduction, from the first we do more than compute. We do linear systems quickly but completely, including the proofs needed to justify what we are computing. Then, with the linear systems work as motivation and at a point where the study of linear combinations seems natural, the second chapter starts with the definition of a real vector space. This occurs by the end of the third week.

Another example of our emphasis on motivation and naturalness is that the third chapter on linear maps does not begin with the definition of homomorphism, but with that of isomorphism. That's because this definition is easily motivated by the observation that some spaces are "just like" others. After that, the next section takes the reasonable step of defining homomorphism by isolating the operation-preservation idea. This approach loses mathematical slickness, but it is a good trade because it comes in return for a large gain in sensibility to students.

One aim of a developmental approach is that students should feel throughout the presentation that they can see how the ideas arise, and perhaps picture themselves doing the same type of work.

The clearest example of the developmental approach taken here—and the feature that most recommends this book—is the exercises. A student progresses most while doing the exercises, so they have been selected with great care. Each problem set ranges from simple checks to reasonably involved proofs. Since an instructor usually assigns about a dozen exercises after each lecture, each section ends with about twice that many, thereby providing a selection. There are even a few problems that are challenging puzzles taken from various journals, competitions, or problems collections. (These are marked with a "?" and as part of the fun, the original wording has been retained as much as possible.) In total, the exercises are aimed to both build an ability at, and help students experience the pleasure of, doing mathematics.

#### Applications and Computers 应用程序和计算机

The point of view taken here, that linear algebra is about vector spaces and linear maps, is not taken to the complete exclusion of others. Applications and the role of the computer are important and vital aspects of the subject. Consequently, each of this book's chapters closes with a few application or computer-related topics. Some are: network flows, the speed and accuracy of computer linear reductions, Leontief Input/Output analysis, dimensional analysis, Markov chains, voting paradoxes, analytic projective geometry, and difference equations.

These topics are brief enough to be done in a day's class or to be given as independent projects for individuals or small groups. Most simply give the reader a taste of the subject, discuss how linear algebra comes in, point to some further reading, and give a few exercises. In short, these topics invite readers to see for themselves that linear algebra is a tool that a professional must have.

#### For people reading this book on their own 对于自学这本书的人

This book's emphasis on motivation and development make it a good choice for self-study. But, while a professional instructor can judge what pace and topics suit a class, if you are an independent student then perhaps you would find some advice helpful.

Here are two timetables for a semester. The first focuses on core material.

 week 星期 Monday 礼拜一 Wednesday 礼拜三 Friday 礼拜五 1 One.I.1 One.I.1, 2 One.I.2, 3 2 One.I.3 One.II.1 One.II.2 3 One.III.1, 2 One.III.2 Two.I.1 4 Two.I.2 Two.II Two.III.1 5 Two.III.1, 2 Two.III.2 Exam 6 Two.III.2, 3 Two.III.3 Three.I.1 7 Three.I.2 Three.II.1 Three.II.2 8 Three.II.2 Three.II.2 Three.III.1 9 Three.III.1 Three.III.2 Three.IV.1, 2 10 Three.IV.2, 3, 4 Three.IV.4 Exam 11 Three.IV.4, Three.V.1 Three.V.1, 2 Four.I.1, 2 12 Four.I.3 Four.II Four.II 13 Four.III.1 Five.I Five.II.1 14 Five.II.2 Five.II.3 Review

The second timetable is more ambitious (it supposes that you know One.II, the elements of vectors, usually covered in third semester calculus).

 week 星期 Monday 礼拜一 Wednesday 礼拜三 Friday 礼拜五 1 One.I.1 One.I.2 One.I.3 2 One.I.3 One.III.1, 2 One.III.2 3 Two.I.1 Two.I.2 Two.II 4 Two.III.1 Two.III.2 Two.III.3 5 Two.III.4 Three.I.1 Exam 6 Three.I.2 Three.II.1 Three.II.2 7 Three.III.1 Three.III.2 Three.IV.1, 2 8 Three.IV.2 Three.IV.3 Three.IV.4 9 Three.V.1 Three.V.2 Three.VI.1 10 Three.VI.2 Four.I.1 Exam 11 Four.I.2 Four.I.3 Four.I.4 12 Four.II Four.II, Four.III.1 Four.III.2, 3 13 Five.II.1, 2 Five.II.3 Five.III.1 14 Five.III.2 Five.IV.1, 2 Five.IV.2

To help you make time trade-offs, in the table of contents I have marked subsections as optional if some instructors will pass over them in favor of spending more time elsewhere. You might also try picking one or two topics that appeal to you from the end of each chapter. You'll get more from these if you have access to computer software that can do the big calculations.

The most important advice is: do many exercises. The recommended exercises are labeled throughout. (The answers are available.) You should be aware, however, that few inexperienced people can write correct proofs. Try to find a knowledgeable person to work with you on this.

Finally, if I may, a caution for all students, independent or not: I cannot overemphasize how much the statement that I sometimes hear, "I understand the material, but it's only that I have trouble with the problems" reveals a lack of understanding of what we are up to. Being able to do things with the ideas is their point. The quotes below express this sentiment admirably. They state what I believe is the key to both the beauty and the power of mathematics and the sciences in general, and of linear algebra in particular (I took the liberty of formatting them as poems).

I know of no better tactic

than the illustration of exciting principles
by well-chosen particulars.
--Stephen Jay Gould

--史蒂芬·杰伊·古尔德

If you really wish to learn

then you must mount the machine
and become acquainted with its tricks
by actual trial.
--Wilbur Wright

--威尔伯·赖特

Jim Hefferon
Mathematics, Saint Michael's College
Colchester, Vermont USA 05439
http://joshua.smcvt.edu
2006-May-20

2006年5月20日

Author's Note. Inventing a good exercise, one that enlightens as well as tests, is a creative act, and hard work.

The inventor deserves recognition. But for some reason texts have traditionally not given attributions for questions. I have changed that here where I was sure of the source. I would greatly appreciate hearing from anyone who can help me to correctly attribute others of the questions.

## 第一章 Linear Systems/线性系统

### 第一节

Systems of linear equations are common in science and mathematics. These two examples from high school science (O'Nan 1990) give a sense of how they arise.

The first example is from Physics. Suppose that we are given three objects, one with a mass known to be 2 kg, and are asked to find the unknown masses. Suppose further that experimentation with a meter stick produces these two balances.

Since the sum of magnitudes of the torques of the clockwise forces equal those of the counter clockwise forces (the torque of an object rotating about a fixed origin is the cross product of the force on it and its position vector relative to the origin; gravitational acceleration is uniform we can divide both sides by it). The two balances give this system of two equations.

${\begin{cases}{\begin{array}{rl}40h+15c&=100\\25c&=50+50h\end{array}}\end{cases}}$ Can you finish the solution?

 c = kg h = kg

The second example of a linear system is from Chemistry. We can mix, under controlled conditions, toluene ${\hbox{C}}_{7}{\hbox{H}}_{8}$ and nitric acid ${\hbox{H}}{\hbox{N}}{\hbox{O}}_{3}$ to produce trinitrotoluene ${\hbox{C}}_{7}{\hbox{H}}_{5}{\hbox{O}}_{6}{\hbox{N}}_{3}$ along with the byproduct water (conditions have to be controlled very well, indeed— trinitrotoluene is better known as TNT). In what proportion should those components be mixed? The number of atoms of each element present before the reaction

$x\,{\rm {C}}_{7}{\rm {H}}_{8}\ +\ y\,{\rm {H}}{\rm {N}}{\rm {O}}_{3}\quad \longrightarrow \quad z\,{\rm {C}}_{7}{\rm {H}}_{5}{\rm {O}}_{6}{\rm {N}}_{3}\ +\ w\,{\rm {H}}_{2}{\rm {O}}$ must equal the number present afterward. Applying that principle to the elements C, H, N, and O in turn gives this system.

${\begin{cases}{\begin{array}{rl}7x&=7z\\8x+1y&=5z+2w\\1y&=3z\\3y&=6z+1w\end{array}}\end{cases}}$ Can you balance the equation?

 ${\rm {C}}_{7}{\rm {H}}_{8}\ +\$ ${\rm {H}}{\rm {N}}{\rm {O}}_{3}\quad \longrightarrow \quad$ ${\rm {C}}_{7}{\rm {H}}_{5}{\rm {O}}_{6}{\rm {N}}_{3}\ +\$ ${\rm {H}}_{2}{\rm {O}}$ To finish each of these examples requires solving a system of equations. In each, the equations involve only the first power of the variables. This chapter shows how to solve any such system.