線性代數（英文維基教科書）/Introduction 簡介

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

這本書幫助學生掌握標準的本科線性代數課程的材料。

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.

材料是標準的，因為涵蓋的主題是高斯約化，向量空間，線性映射，行列式，特徵值和特徵向量。聽眾也是標準的：大二或大三，通常至少有一個學期的微積分背景，也許有三個學期的時間。

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.

它給學生的幫助來自於採取一種發展的方法這本書的介紹強調動機和自然性，由各種各樣的例子和廣泛的，仔細的，練習。發展的方法使這本書與眾不同，所以在這裏對這個術語進行一些擴展是合適的。

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.

大多數數學課程開始時的課程對學生的獎勵較少，因為他們理解了理論，而更多的是因為他們正確地應用了公式和算法。以後的課程要求數學成熟：能夠理解不同類型的論據，熟悉許多數學研究的主題，如基本集合和函數事實，以及獨立閱讀和思考的能力。線性代數是研究這兩種課程之間過渡的理想場所。它在一個項目中出現得早，這樣在這裏取得的進步會得到回報，但也會來得太晚，以至於學生通常都是主修生和未成年學生。材料連貫，通俗易懂，優雅大方。用矛盾證明、當且僅當語句證明、歸納證明等多種論證方式，如實例和實例豐富。

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，這本書的目的是幫助學生從目前的水平，在大多數成員主要對科學或工程應用感興趣的課程中取得成功，發展到下一階段的成功，即數學學科本身的嚴肅學生。

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.

幫助學生完成這一轉變意味着要認真對待數學，因此本書中的所有結果都得到了證明。另一方面，我們不能假設學生已經到了，因此與更抽象的文本相比，我們給出了許多例子，而且往往非常詳細。

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.

在過去，線性代數文本通常會突然進行這種轉換。他們開始廣泛計算線性系統，矩陣乘法和行列式。當概念向量空間和線性映射最終出現，定義和證明開始時，這種變化常常使學生停止。在這本書中，我們從一個計算主題開始，線性化簡，從一開始我們做的不僅僅是計算。我們快速但完整地處理線性系統，包括證明我們正在計算的東西。然後，以線性系統為動力，在研究線性組合似乎很自然的地方，第二章從實向量空間的定義開始。這將在第三周結束時發生。

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.

我們強調動機和自然性的另一個例子是，關於線性映射的第三章沒有從同態的定義開始，而是從同構的定義開始。這是因為這個定義很容易被一些空間「和」其他空間「一樣」的觀察所激發。然後，下一節通過隔離操作保持的思想，採取合理的步驟來定義同態。這種方法失去了數學上的圓滑，但它是一種很好的交易，因為它可以讓學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.

發展性教學法的一個目的是讓學生在整個演示過程中感覺到他們可以看到想法是如何產生的，也許還能想像自己在做同樣類型的工作。

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，在這裏採取的發展方法和特點，最推薦這本書是練習。學生在做練習時進步最大，所以他們是經過精心挑選的。每個習題集的範圍從簡單的檢查到合理涉及的證明。由於教師通常在每堂課後佈置十幾個練習題，所以每節課結束時的練習數是原來的兩倍，因此提供了一個選擇題。甚至有一些問題是挑戰性的難題從各種雜誌，比賽，或問題收集。（這些標記有「？」總的來說，這些練習的目的是培養學生學習數學的能力，並幫助他們體驗數學的樂趣。

Applications and Computers
應用程式和計算機

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.

這裏所採取的觀點，即線性代數是關於向量空間和線性映射的，並不完全排除其他的觀點。計算機的應用和作用是這門學科重要而重要的方面。因此，本書的每一章都以一些應用或計算機相關的主題結束。其中包括：網絡流、計算機線性化簡的速度和精度、Leontief輸入/輸出分析、量綱分析、馬爾可夫鏈、投票悖論、解析射影幾何和差分方程。

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.

這些主題足夠簡短，可以在一天的課堂上完成，也可以作為個人或小組的獨立項目。最簡單的是讓讀者領略一下這個主題，討論一下線性代數是如何產生的，指出一些進一步的閱讀，並給出一些練習。簡言之，這些主題邀請讀者親眼看到，線性代數是一個專業人士必須具備的工具。

For people reading this book on their own
對於自學這本書的人

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.

這本書對動機和發展的強調使它成為自學的好選擇。但是，雖然專業的教師可以判斷什麼樣的節奏和主題適合一節課，但如果你是一名獨立學生，那麼也許你會發現一些建議是有用的。

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

這是一個學期的兩個時間表。第一個重點是核心材料。

week 星期	Monday 禮拜一	Wednesday 禮拜三	Friday 禮拜五
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).

第二個時間表更具雄心（它假設你知道一、二，向量的元素，通常在第三學期微積分中討論）。

week 星期	Monday 禮拜一	Wednesday 禮拜三	Friday 禮拜五
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.

這些小節的標題見目錄。

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.

為了幫助您進行時間權衡，在目錄中，我將子部分標記為可選的，如果有些講師會跳過它們，而將更多的時間花在其他地方。你也可以試着從每一章的結尾選一兩個吸引你的話題。如果你能使用計算機軟件進行大計算，你會從中得到更多。

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.

最重要的建議是：多做練習。推薦的練習貫穿始終。（答案是有的）但是你應該知道，沒有經驗的人很少能寫出正確的證明。試着找一個有見識的人和你一起工作。

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).

最後，如果可以的話，我要提醒所有的學生，不管他們是否獨立：我不能過分強調我有時聽到的「我理解材料，但只是我對問題有困難」這句話多少暴露了我們對我們正在做的事情缺乏了解。能夠用這些想法做事是他們的重點。下面的引文很好地表達了這種觀點。它們陳述了我認為是數學和科學的美和力量的關鍵，尤其是線性代數（我冒昧地將它們格式化為詩歌）。

I know of no better tactic

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

我知道沒有比這更好的策略了
而不是那些令人興奮的原則
通過精心挑選的細節。
--史蒂芬·傑伊·古爾德

If you really wish to learn

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

如果你真的想學
那你就得裝上機器
熟悉它的技巧
通過實際試驗。
--威爾伯·賴特

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

吉姆·赫弗倫 聖米歇爾學院數學系 美國佛蒙特州科爾切斯特市，郵編：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.

作者的筆記。創造一個好的練習，一個啟發和測試，是一個創造性的行為，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.

這位發明家值得肯定。但由於某些原因，文本傳統上沒有給出問題的歸屬。我已經改變了這裏我確定來源的地方。如果有人能幫助我正確回答其他問題，我將不勝感激。