线性代数（英文维基教科书）/Introduction 简介

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

这本书帮助学生掌握标准的本科线性代数课程的材料。

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.

材料是标准的，因为涵盖的主题是高斯约化，向量空间，线性映射，行列式，特征值和特征向量。听众也是标准的：大二或大三，通常至少有一个学期的微积分背景，也许有三个学期的时间。

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.

它给学生的帮助来自于采取一种发展的方法这本书的介绍强调动机和自然性，由各种各样的例子和广泛的，仔细的，练习。发展的方法使这本书与众不同，所以在这里对这个术语进行一些扩展是合适的。

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.

大多数数学课程开始时的课程对学生的奖励较少，因为他们理解了理论，而更多的是因为他们正确地应用了公式和算法。以后的课程要求数学成熟：能够理解不同类型的论据，熟悉许多数学研究的主题，如基本集合和函数事实，以及独立阅读和思考的能力。线性代数是研究这两种课程之间过渡的理想场所。它在一个项目中出现得早，这样在这里取得的进步会得到回报，但也会来得太晚，以至于学生通常都是主修生和未成年学生。材料连贯，通俗易懂，优雅大方。用矛盾证明、当且仅当语句证明、归纳证明等多种论证方式，如实例和实例丰富。

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.

因此，这本书的目的是帮助学生从目前的水平，在大多数成员主要对科学或工程应用感兴趣的课程中取得成功，发展到下一阶段的成功，即数学学科本身的严肃学生。

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.

帮助学生完成这一转变意味着要认真对待数学，因此本书中的所有结果都得到了证明。另一方面，我们不能假设学生已经到了，因此与更抽象的文本相比，我们给出了许多例子，而且往往非常详细。

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.

在过去，线性代数文本通常会突然进行这种转换。他们开始广泛计算线性系统，矩阵乘法和行列式。当概念向量空间和线性映射最终出现，定义和证明开始时，这种变化常常使学生停止。在这本书中，我们从一个计算主题开始，线性化简，从一开始我们做的不仅仅是计算。我们快速但完整地处理线性系统，包括证明我们正在计算的东西。然后，以线性系统为动力，在研究线性组合似乎很自然的地方，第二章从实向量空间的定义开始。这将在第三周结束时发生。

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.

我们强调动机和自然性的另一个例子是，关于线性映射的第三章没有从同态的定义开始，而是从同构的定义开始。这是因为这个定义很容易被一些空间“和”其他空间“一样”的观察所激发。然后，下一节通过隔离操作保持的思想，采取合理的步骤来定义同态。这种方法失去了数学上的圆滑，但它是一种很好的交易，因为它可以让学生在情感上得到很大的提高。

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.

发展性教学法的一个目的是让学生在整个演示过程中感觉到他们可以看到想法是如何产生的，也许还能想象自己在做同样类型的工作。

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.

最清楚的例子，在这里采取的发展方法和特点，最推荐这本书是练习。学生在做练习时进步最大，所以他们是经过精心挑选的。每个习题集的范围从简单的检查到合理涉及的证明。由于教师通常在每堂课后布置十几个练习题，所以每节课结束时的练习数是原来的两倍，因此提供了一个选择题。甚至有一些问题是挑战性的难题从各种杂志，比赛，或问题收集。（这些标记有“？”总的来说，这些练习的目的是培养学生学习数学的能力，并帮助他们体验数学的乐趣。

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.

这里所采取的观点，即线性代数是关于向量空间和线性映射的，并不完全排除其他的观点。计算机的应用和作用是这门学科重要而重要的方面。因此，本书的每一章都以一些应用或计算机相关的主题结束。其中包括：网络流、计算机线性化简的速度和精度、Leontief输入/输出分析、量纲分析、马尔可夫链、投票悖论、解析射影几何和差分方程。

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.

这些主题足够简短，可以在一天的课堂上完成，也可以作为个人或小组的独立项目。最简单的是让读者领略一下这个主题，讨论一下线性代数是如何产生的，指出一些进一步的阅读，并给出一些练习。简言之，这些主题邀请读者亲眼看到，线性代数是一个专业人士必须具备的工具。

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.

这本书对动机和发展的强调使它成为自学的好选择。但是，虽然专业的教师可以判断什么样的节奏和主题适合一节课，但如果你是一名独立学生，那么也许你会发现一些建议是有用的。

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

这是一个学期的两个时间表。第一个重点是核心材料。

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).

第二个时间表更具雄心（它假设你知道一、二，向量的元素，通常在第三学期微积分中讨论）。

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

See the table of contents for the titles of these subsections.

这些小节的标题见目录。

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.

为了帮助您进行时间权衡，在目录中，我将子部分标记为可选的，如果有些讲师会跳过它们，而将更多的时间花在其他地方。你也可以试着从每一章的结尾选一两个吸引你的话题。如果你能使用计算机软件进行大计算，你会从中得到更多。

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.

最重要的建议是：多做练习。推荐的练习贯穿始终。（答案是有的）但是你应该知道，没有经验的人很少能写出正确的证明。试着找一个有见识的人和你一起工作。

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).

最后，如果可以的话，我要提醒所有的学生，不管他们是否独立：我不能过分强调我有时听到的“我理解材料，但只是我对问题有困难”这句话多少暴露了我们对我们正在做的事情缺乏了解。能够用这些想法做事是他们的重点。下面的引文很好地表达了这种观点。它们陈述了我认为是数学和科学的美和力量的关键，尤其是线性代数（我冒昧地将它们格式化为诗歌）。

I know of no better tactic

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

我知道没有比这更好的策略了
而不是那些令人兴奋的原则
通过精心挑选的细节。
--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

如果你真的想学
那你就得装上机器
熟悉它的技巧
通过实际试验。
--威尔伯·赖特

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

吉姆·赫弗伦 圣米歇尔学院数学系 美国佛蒙特州科尔切斯特市，邮编：05439

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.

作者的笔记。创造一个好的练习，一个启发和测试，是一个创造性的行为，努力工作。

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.

这位发明家值得肯定。但由于某些原因，文本传统上没有给出问题的归属。我已经改变了这里我确定来源的地方。如果有人能帮助我正确回答其他问题，我将不胜感激。