Andrilli & Hecker's Elementary Linear Algebra, 4th Edition: A Comprehensive Guide

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"Elementary Linear Algebra, 4ed(Andrilli, Hecker)" 《Elementary Linear Algebra》第四版是由Stephen Andrilli和David Hecker合著的一本基础线性代数教材，适合大学数学和计算机科学专业学生使用。该书由Academic Press（Elsevier的一个印记）出版，旨在为读者提供线性代数的基本概念、理论和应用。线性代数是数学的一个重要分支，它研究向量、矩阵、线性方程组、线性变换以及这些概念的几何表示。这本书可能涵盖了以下主要知识点： 1. **向量和向量空间**：向量作为线性代数的基础，描述了大小和方向。向量空间则定义了一组向量的集合，其中包含加法、标量乘法等运算规则。 2. **线性组合与线性独立**：理解向量可以如何通过标量的加权组合形成新的向量，以及一组向量是否可以由另一组更小的向量唯一确定。 3. **基和坐标**：在给定的向量空间中，基是一组线性独立的向量，可以用来表示空间中的所有其他向量。坐标是将向量表示为基向量的线性组合时的标量系数。 4. **矩阵**：矩阵是二维数组，用于表示线性变换或系统线性方程。矩阵的运算包括加法、乘法和标量乘法。 5. **行列式**：对于方阵，行列式提供了关于其逆和秩的信息。行列式为零意味着矩阵不可逆，对应线性方程组无解或多解。 6. **特征值与特征向量**：线性变换的特征值和特征向量揭示了变换对特定向量的作用，它们在物理学、工程学和其他领域中有广泛应用。 7. **线性方程组的解**：通过高斯消元法、矩阵方法（如克拉默法则）来求解线性方程组。 8. **欧几里得空间与范数**：引入了距离和角度的概念，使线性代数具有几何意义。范数是衡量向量长度的标准。 9. **正交性和正交投影**：正交向量组和正交矩阵在处理物理问题和信号处理中非常关键，正交投影是将向量投射到另一个向量或子空间上的过程。 10. **特征值问题和二次型**：讨论二次型的标准化和对称矩阵，它们与二次曲面和最优化问题有关。《Elementary Linear Algebra》第四版可能还包含了各种实际应用示例，如电路分析、图像处理和物理学中的问题，以帮助学生更好地理解和应用这些理论知识。此外，书中的习题和练习有助于巩固学习，并提供了解决实际问题的练习机会。

Preface for the Instructor xv

Section Prerequisite

Section 8.8 (Computer Graphics) Section 5.2 (The Matrix of a Linear Transformation)

Section 8.9 (Differential Equations)** Section 5.6 (Diagonalization of Linear Operators)

Section 8.10 (Least-Squares Section 6.2 (Orthogonal Complements)

Solutions for Inconsistent Systems)

Section 8.11 (Quadratic Forms) Section 6.3 (Orthogonal Diagonalization)

Section 9.1 (Numerical Methods for Section 2.3 (Equivalent Systems, Rank,

Solving Systems) and Row Space)

Section 9.2 (LDU Decomposition) Section 2.4 (Inverses of Matrices)

Section 9.3 (The Power Method Section 3.4 (Eigenvalues and Diagonalization)

for Finding Eigenvalues)

Section 9.4 (QR Factorization) Section 6.1 (Orthogonal Bases and the Gram-Schmidt

Process)

Section 9.5 (Singular Value Section 6.3 (Orthogonal Diagonalization)

Decomposition)

(Continued)

*In addition to the prerequisites listed, each section in Chapter 7 requires the sections of Chapter 7 that precede

it, although most of Section 7.5 can be covered without having covered Sections 7.1 through 7.4 by concentrating

only on real inner products.

**The techniques presented for solving differential equations in Section 8.9 require only Section 3.4 as a

prerequisite. However, terminology from Chapters 4 and 5 is used throughout Section 8.9.

PLANS FOR COVERAGE

Chapters 1 through 6 have been written in a sequential fashion. Each section is gen-

erally needed as a prerequisite for what follows.Therefore, we recommend that these

sections be covered in order. However, there are three exceptions:

■ Section 1.3 (An Introduction to Proofs) can be covered,in whole, or in part,

at any time after Section 1.2.

■ Section 3.3 (Further Properties of the Determinant) contains some material

that can be omitted without affecting most of the remaining development. The

topics of general cofactor expansion,(classical) adjoint matrix,and Cramer’s Rule

are used very sparingly in the rest of the text.

■ Section 6.1 (Orthogonal Bases and the Gram-Schmidt Process) can be

covered any time after Chapter 4, as can much of the material in Section 6.2

(Orthogonal Complements).

Any section in Chapters 7 through 9 can be covered at any time as long as the

prerequisites for that section have previously been covered. (Consult the Prerequisite

Chart for Sections in Chapters 7, 8, 9.)

Preface for the Student

OVERVIEW OF THE MATERIAL

Chapters 1 to 3: Appetizer: Linear algebra is a branch of mathematics that is largely

concerned with solving systems of linear equations. The main tools for working with

systems of linear equations are vectors and matrices.Therefore,this text begins with an

introduction to vectors and matrices and their fundamental properties in Chapter 1.

This is followed by techniques for solving linear systems in Chapter 2. Chapter 3

introduces determinants and eigenvalues, which help us to better understand the

behavior of linear systems.

Chapters 4 to 7: Main Course: The material of Chapters 1, 2, and 3 is treated in a more

abstract form in Chapters 4 through 7. In Chapter 4, the concept of a vector space

(a collection of general vectors) is introduced, and in Chapter 5, mappings between

vector spaces are considered. Chapter 6 explores orthogonality in the most common

vector space, and Chapter 7 considers more general types of vector spaces, such as

complex vector spaces and inner product spaces.

Chapters 8 and 9: Dessert: The powerful techniques of linear algebra lend themselves

to many important and diverse applications in science, social science, and business,

as well as in other branches of mathematics. While some of these applications are

covered in the text as new material is introduced, others of a more lengthy nature are

placed in Chapter 8, which is entirely devoted to applications of linear algebra. There

are also many useful numerical algorithms and methods associated with linear algebra,

some of which are covered in Chapters 1 through 7. Additional numerical algorithms

are explored in Chapter 9.

HELPFUL ADVICE

Strategies for Learning: Many students ﬁnd the transition to abstractness that begins

in Chapter 4 to be challenging. This textbook was wr itten speciﬁcally to help you in

this regard. We have tried to present the material in the clearest possible manner with

many helpful examples.We urge you to take advantage of this and read each section

of the textbook thoroughly and carefully many times over. Each re-reading will allow

you to see connections among the concepts on a deeper level. Try as many problems

in each section as possible. There are True/False questions to test your knowledge at

the end of each section, as well as at the end of each of the sets of Review Exercises

for Chapters 1 to 7. After pondering these ﬁrst on your own, consult the explanations

for the answers in the Student Solutions Manual.

Facility with Proofs: Linear algebra is considered by many instructors as a transi-

tional course from the freshman computationally-oriented calculus sequence to the

xix

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