英文版线性代数经典教材：Elementary Linear Algebra

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"Elementary Linear Algebra" 线性代数是数学的一个基础分支，主要研究向量、矩阵、线性变换以及它们在几何、代数和应用科学中的作用。"Elementary Linear Algebra" 是一本经典的线性代数教科书，提供高清版本，适用于初学者和对线性代数有基本需求的学生。该书可能包含了以下核心知识点： 1. **向量与向量空间**：向量是线性代数的基本元素，可以表示在二维或三维空间中的箭头。向量空间是一组向量的集合，满足加法和标量乘法的封闭性。向量空间中的操作包括向量的加法、减法和标量乘法。 2. **矩阵**：矩阵是由有序数组构成的矩形阵列，用于表示线性方程组、线性变换等。矩阵的运算包括加法、减法、标量乘法和矩阵乘法。 3. **线性组合与线性独立**：一组向量可以通过标量的线性组合来表示另一组向量。如果一个向量集合不能被其他任何更小的集合通过线性组合表示，那么它们是线性独立的。 4. **行列式**：对于方阵（行数和列数相等的矩阵），行列式是一个标量值，它可以用来判断矩阵是否可逆，以及矩阵所代表的线性变换是否会改变面积或体积。 5. **秩与零度**：矩阵的秩是指其行向量或列向量生成的空间的维数，反映了矩阵的“线性复杂度”。零度是矩阵中非零元素的最小数量，零度等于秩时，矩阵是满秩的。 6. **线性方程组的解**：线性代数研究如何求解线性方程组。高斯消元法、克拉默法则和矩阵的逆都是求解线性方程组的方法。 7. **特征值与特征向量**：对于一个给定的方阵，特征值和特征向量揭示了矩阵作为线性变换的本质。特征值是满足特定关系的标量，而特征向量是与这些标量相对应的向量。 8. **二次型与对称矩阵**：二次型是二阶多项式的形式，通常与对称矩阵相关联。通过对角化或正交变换，可以将二次型转化为标准形式。 9. **线性变换**：线性变换是将一个向量空间映射到另一个向量空间的规则，保持向量的线性性质。线性变换可以用矩阵表示，并且其性质可以通过矩阵的运算来研究。 10. **内积与欧几里得空间**：内积是定义在向量空间上的一种二元运算，它给出了向量之间的“角度”和“长度”。在内积空间，特别是欧几里得空间中，可以定义向量的长度和向量之间的角度。 "Elementary Linear Algebra" 可能会详细讲解这些概念，并通过实例和习题帮助读者理解和掌握线性代数的基本原理及其应用。书中还可能包含使用计算机软件（如MATLAB）解决线性代数问题的指导，以增强理论学习的实际应用能力。

Preface

In beginning to write a linear algebra text, a question that surfaces even before the rst

keystroke takes place is who is the audience and what do we want to accomplish. The

answer to this is more complex than in several other areas of mathematics because of the

breadth of potential users. The book was written with the idea that a typical student would

be one who has completed two semesters of calculus but who has not taken courses that

emphasize abstract mathematics. The goals of the text are to present the topics that are

traditionally covered in a rst-level linear algebra course so that the computational meth-

ods become deeply ingrained and to make the intuition of the theory as transparent as

possible.

Many disciplines, including statistics, economics, environmental science, engineering,

and computer science, use linear algebra extensively. The sophistication of the applications

of linear algebra in these areas can vary greatly. Students intending to study mathematics

at the graduate level, and many others, would benet from having a second course in lin-

ear algebra at the undergraduate level.

Some of the computations that we feel are especially important are matrix computations,

solving systems of linear equations, representing a linear transformation in standard bases,

nding eigenvectors, and diagonalizing matrices. Of less emphasis are topics such as con-

verting the representation of vectors and linear transformations between nonstandard

bases and converting a set of vectors to a basis by expanding or contracting the set.

In some cases, the intuition of a proof is more transparent if an example is presented

before a theorem is articulated or if the proof of a theorem is given using concrete cases

rather than an abstract argument. For example, by using three vectors instead of

nvectors.

There are places in Chapters 4 through 7 where there are results that are important

because of their applications, but the theory behind the result is time consuming and is

more advanced than a typical student in a rst exposure would be expected to digest. In

such cases, the reader is alerted that the result is given later in the section and omitting the

derivation will not compromise the usefulness of the results. Two specic examples of this

are the projection matrix and the Gram–Schmidt process.

The exercises were designed to span a range from simple computations to fairly direct

abstract exercises.

We expect that most users will want to make extensive use of a computer algebra system

for computations. While there are several systems available, MATLAB

is the choice of

many, and we have included a tutorial for MATLAB in the appendix. Because of the exten-

sive use of the program R by statisticians, a tutorial for that program is also included.

A Note about Mathematical Proofs

As a text for a rst course in linear algebra, this book has a major focus on demonstrating

facts and techniques of linear systems that will be invaluable in higher mathematics

and elds that use higher mathematics. This entails developing many computational tools.

xvi Preface

A rst course in linear algebra also serves as a bridge to mathematics courses that are pri-

marily theoretical in nature and, as such, necessitates understanding and, in some cases,

developing mathematical proofs. This is a learning process that can be frustrating.

We offer some suggestions that may ease the transition.

First, the most important requirement for abstract mathematics is knowing the impor-

tant denitions well. These are analogous to the formulas of computational mathematics.

A casual perusal will not be sufcient.

You will often encounter the question “Is it true that …?” Here you must show that a

result holds in all cases allowed by the hypotheses or nd one example where the claim

does not hold. Such an example is called a counter example. If you are going to prove that

a result always holds, it is helpful (perhaps necessary) to believe that it holds. One way to

establish an opinion for its validity is to work through some examples. This may highlight

the essential properties that are crucial to a proof. If the result is not valid, this will often

yield a counter example. Simple examples are usually the best. In linear algebra, this usu-

ally means working with small dimensional objects, such as 2 × 2 matrices.

In the text, we will sometimes give the ideas of a proof rather than a complete proof

when we feel that “what is really going on” is obscured by the abstraction, for example, if

the idea of a proof seems clear by examining 2 × 2 matrices but would be difcult to follow

forn × n matrices.

In constructing a mathematical proof, there are two processes involved. One is to develop

the ideas of the proof, and the second is to present those ideas in appropriate language.

There are different methods of proof that can be broadly divided into direct methods and

indirect methods. An example of an indirect method is proof by contradiction. The contra-

positive of “if statement A is true, then statement B is true” is “if statement B is false, then

statement A is false.” Proving the contrapositive of an “if-then” statement is logically

equivalent to proving the original statement. The contrapositive of

“”If ,thenisanintegerxx= 4

“.”If is notaninteger,thenxx¹ 4

The converse of an “if-then” statement reverses the hypothesis and the conclusion. The

converse of “if statement A is true, then statement B is true” is “ if statement B is true, then

statement A is true.”

Unlike the contrapositive, the converse is not logically equivalent to the original state-

ment. The converse of

“, ”If then is an integerxx= 4

“,.”If is an integer thenxx= 4

Here the original statement is true, but the converse is false. You will sometimes have a

problem of the type “show ‘A if and only if B.’” In this case, you must prove

“”If A, then B

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