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| 內容簡介： |
本书叙述深入浅出，以矩阵为主线，突出矩阵的运算和化简，突出用矩阵方法研究线性方程组、二次型和实际问题模型。本书对于抽象的理论和方法，总是从具体问题入手，再将其推广到一般情形，而略去了许多繁杂的理论推导，并力求将数学与应用相结合。
本书的主要内容包括线性方程组、矩阵代数、行列式、向量空间、矩阵的特征值与特征向量和二次型等。
本书是一本介绍性的线性代数教材，内容简洁，层次清晰，适合高等学校理工科专业线性代数课程双语教学使用。
本书叙述深入浅出，以矩阵为主线，突出矩阵的运算和化简，突出用矩阵方法研究线性方程组、二次型和实际问题模型。本书对于抽象的理论和方法，总是从具体问题入手，再将其推广到一般情形，而略去了许多繁杂的理论推导，并力求将数学与应用相结合。
本书的主要内容包括线性方程组、矩阵代数、行列式、向量空间、矩阵的特征值与特征向量和二次型等。
本书是一本介绍性的线性代数教材，内容简洁，层次清晰，适合高等学校理工科专业线性代数课程双语教学使用。
The matrix is the mainline of the book. With the help of the matrix operation
and the matrix simplification, we study the linear equations, the quadratic
forms and the real world applications. For the purpose of the insights into the
abstract theory and the methods of the linear algebra, we start to discuss the
conceptions and the methods with the specific problems, then we directly extend
them to the general situation without the complicated theoretical derivation.
Furthermore, we try to combine the mathematical methods with the real
applications in this book.
The main contents of the book are linear equations, matrix algebra,
determinants, vector spaces, eigenvalues and eigenvectors，and quadratic forms, etc.
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| 目錄：
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Chapter 1 Linear Equations in Linear
Algebra001
1.1Systems of Linear Equations001
1.2Row Reduction and Echelon Forms008
1.3Solutions of Linear Systems012
1.4Vector Equations014
Exercises017
Chapter 2 Matrix Algebra019
2.1Matrix Operations019
2.2The Inverse of a Matrix024
2.3Partitoned Matrices028
2.4Matrix Factorizations031
2.5Subspace of Rn032
2.6Dimension and Rank035
Exercises037
Chapter 3 Determinants040
3.1Introduction to Determinants040
3.2Properties of Determinants043
3.3Cofactor Expansion048
3.4The Inverse of a Matrix050
3.5Cramers Rule053
Exercises054
Chapter 4 Vector Spaces058
4.1Definition of Vector Spaces058
4.2Subspaces and Span062
4.3Linearly Independent Sets068
4.4Bases and Dimension071
4.5Inner Product，Length，Angle074
4.6Orthonormal Basis and the Gram-Schmidt Procedure078
Exercises084
Chapter 5 Eigenvalues and Eigenvectors088
5.1Definition of Eigenvalues and Eigenvectors088
5.2Properties of Eigenvalues and Eigenvectors092
5.3Similarity and Diagonalization096
5.4Diagonalization of Symmetric Matrices100
Exercises105
Chapter 6 Solution Sets of Linear Systems107
6.1Homogeneous Linear Systems107
6.2Solutions of Nonhomogeneous Systems108
6.3Applications of Linear Systems110
Exercises113
Chapter 7 Symmetric Matrices and Quadratic Forms117
7.1Diagonalization of Symmetric Matrices117
7.2Quadratic Forms119
7.3Quadratic Problems122
7.4The Singular Value Decomposition126
7.5Applications to Statistics129
Exercises132
References134
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| 內容試閱：
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The main goal of the text is to help students master the
basic concepts and skills they will use later in their study and careers.The
text provides a modern elementary introduction to linear algebra and a broad
selection of interesting applications.The material is accessible to students
with the maturity that should come from successful completion of two semesters
of college-level calculus.
We have attempted to give this book the following distinctive features.
（1）Many
fundamental ideas of linear algebra are introduced within the first lectures，then gradually examined from different points of view.A major
achievement of the text is that the level of difficulty is fairly even.
（2）Good notation is crucial，and the text reflects the way scientists and engineers actually use
linear algebra in practice.The definitions and proofs focus on the columns of a
matrix rather than on the matrix entries.This modern approach simplifies many
arguments，and it ties vector space ideas into the study
of linear systems.
（3）A broad selection of
applications illustrates the power of linear algebra to explain fundamental
principles and simplify computing in engineering，physics，economics，and statistics.
In this volume，Chapter 1，Chapter
2，Chapter 6 and Chapter 7 are written by Professor Guoqing
Liu，Chapter 3 is written by Associated Professor Jian
Zhao，and Chapter 4 and Chapter 5 are written by Dr.Wei
Shi.All the chapters are checked and revised by Professor Guoqing Liu.
We hope this book can bring readers some help in the studying and teaching of
bilingual mathematics.Due to the limit of our ability，it
is impossible to avoid some unclear explanations.We would appreciate any
constructive criticisms and corrections from readers.
Guoqing Liu，Jian Zhao，Wei Shi
2019.6
本书的主要目标是帮助学生们掌握他们在后面的课程学习和未来事业发展中需要的一些线性代数的基本原理和相关技能。本书以现代的视野简明扼要地介绍了线性代数的基本思想和具有广泛背景的有趣应用。这些内容的学习需要学生们具备两个学期的大学微积分课程基础。
我们致力于使本书具备以下特征。
（1）线性代数涉及的许多基本思想和概念在开始的几讲中通过简单的案例引入，再通过随后章节的反复论述，从不同的角度逐渐强化对这些思想和概念的理解。本书的一大成果是通过层层递进的叙述方式降低了对线性代数概念理解的难度。
（2）本书用科学家和工程师们在实际问题中应用线性代数时所熟悉的方法和符号来描绘线性代数。例如，很多的定义和证明都是针对矩阵的列而不是矩阵的元素进行的，使得这些定义和证明看上去更简洁。同时，这样做也将向量空间的思想揉入线性代数方程组的研究中，使得方程组有关解的结论更具应用价值。
（3）书中广泛选择的应用问题彰显了线性代数在解释经济、统计和工程等领域的基本原理方面的重要性，同时也表明利用线性代数可以简化这些领域的科学计算。
本书的第一章、第二章、第六章和第七章由刘国庆教授撰写，第三章由赵剑副教授撰写，第四章和第五章由石玮副教授撰写。全书由刘国庆教授统稿。
我们希望本书的出版对那些用双语学习和教授线性代数的读者有所帮助。由于我们能力的局限，在解释有关概念、理论和方法时，不可避免地会出现疏漏。我们真诚地希望得到来自读者的批评和指正。
本书的编写得到江苏省第二批中外合作办学高水平示范性建设工程项目培育点：南京工业大学与英国谢菲尔德大学合作举办数学与应用数学（金融数学）专业 本科教育项目（苏教办外【2017】14号）经费支持。
刘国庆，赵剑，石玮
2019年6月
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