基本信息
- 原书名：Linear Algebra:An Interative Approach
- 原出版社： Thomson
- 作者： [美]S.K.Jain,A.D.Gunawardena
- 丛书名： 经典原版书库
- 出版社：机械工业出版社
- ISBN：9787111125112
- 上架时间：2003-8-14
- 出版日期：2003 年7月
- 开本：16开
- 页码：418
- 版次：1-1
- 所属分类：数学 > 代数，数论及组合理论 > 线性代数
教材 > 研究生/本科/专科教材 > 理学 > 数学
内容简介
作译者
S．K．Jain, 俄亥俄大学数学系教授，同时，他还是加利福尼亚大学伯克利分校、俄亥俄州立大学、北卡罗来纳州立大学、芝加哥大学及印度理工学院等多所著名大学的客座教授。他在专业杂志上发表过100余篇论文，并著有4本本科生／研究生教材。
A．D.Gunawardena，计算机科学硕士、应用数学博士，现在卡内基·梅隆大学计算机科学系任教，目前的研究方向为数据挖掘应用和构建智能学习系统的人机交互原理。
A．D.Gunawardena，计算机科学硕士、应用数学博士，现在卡内基·梅隆大学计算机科学系任教，目前的研究方向为数据挖掘应用和构建智能学习系统的人机交互原理。
目录
1 Linear Systems and Matrices 1
1.1 Linear Systems of Equations 1
1.2 Elementary Operations and Gauss Elimination Method
1.3 Homogeneous Linear Systems 11
1.4 Introduction to Matrices and the Matrix of a Linear System 16
1.5 Elementary Row Operations on a Matrix 21
1.6 Proofs of Facts 28
1.7 Chapter Review Questions and Project 29
2 Algebra of Matrices 33
2.1 Scalar Multiplication and Addition of Matrices 33
2.2 Matrix Multiplication and Its Properties 38
2.3 Transpose 49
2.4 Proofs of Facts 52
2.5 Chapter Review Questions and Projects 55
3 Subspaces 61
3.1 Linear Combination of Vectors 61
3.2 Vector Subspaces 66
3.3 Linear Dependence, Linear Independence, and Basis 75
3.4 Proofs of Facts 84
3.5 Chapter Review Questions and Project 87
1.1 Linear Systems of Equations 1
1.2 Elementary Operations and Gauss Elimination Method
1.3 Homogeneous Linear Systems 11
1.4 Introduction to Matrices and the Matrix of a Linear System 16
1.5 Elementary Row Operations on a Matrix 21
1.6 Proofs of Facts 28
1.7 Chapter Review Questions and Project 29
2 Algebra of Matrices 33
2.1 Scalar Multiplication and Addition of Matrices 33
2.2 Matrix Multiplication and Its Properties 38
2.3 Transpose 49
2.4 Proofs of Facts 52
2.5 Chapter Review Questions and Projects 55
3 Subspaces 61
3.1 Linear Combination of Vectors 61
3.2 Vector Subspaces 66
3.3 Linear Dependence, Linear Independence, and Basis 75
3.4 Proofs of Facts 84
3.5 Chapter Review Questions and Project 87
前言
Introduction
The purpose of this book is to provide an introduction to linear algebra, a branch of mathematics dealing with matrices and vector spaces. Matrices have been introduced here as a handy tool for solving systems of linear equations. But their utility goes far beyond this initial application. There is hardly any area of modem mathematics in which matrices do not have some application. They have also many applications in other disciplines, such as statistics, economics, engineering, physics, chemistry, biology, geology, and business.
The present text in linear algebra is designed for a general audience of sophomore-level students majoring in any area of art, science, or engineering. The only prerequisites are two or three years of high school mathematics with some knowledge of calculus.
A special feature of this book is that it can be used in a course taught in a traditional
manner as well as in a course using technology. Those using technology may refer to complete solutions of selected exercises (marked as drills) using Matlab at the end of the book, while others using Maple or Mathematica may refer to corresponding solutions on the web page. The readers would find the examples and solutions to drills using technology quite helpful and illustrative to solve similar problems. Concepts and practical methods for solving problems are illustrated through plenty of examples. The theorems and facts underlying these methods are clearly stated as they arise, but their proofs are provided in a separate section called "Proofs of Facts" near the end of thechapter.
Text Organization
The subject matter is laid out ill a leisurely manner with plenty of examples to illus-trate concepts and applications. Most of the sections contain a fairly large number of exercises, some of which relate to real-life problems. Chapters 1 and 2 deal with linear systems, leading naturally to matrices and to algebraic operations on matrices.
Chapter 3 introduces the vector space F" of n-tuples, linear dependence and indepen-dence of vectors. The concept of rank is introduced in Chapter 4 and is followed by more applications of elementary row operations in Chapter 5. Specifically, we show in Chapter 5 how to find the inverse of an invertible matrix, the LU-decomposition and full rank factorization of a matrix. Chapter 6 provides a working knowledge of determinants, which are later considered in a rigorous fashion in Chapter 10. Eigen-value problems and inner product spaces are given in Chapters 7 and 8, respectively.
An interesting feature of Chapter 7 is a method for finding eigenvalues without determinants. Two methods for finding the least-squares solution of an inconsistent linear system are given in Chapter 8. The first method is geometric and uses the notion of shortest distance; the second is algebraic and uses the concept of generalized in-verses. Vector spaces are revisited in Chapter 9, where a formal definition of a vector space is given and important examples of vector spaces of functions are considered.
In addition to the answers to all exercises, we have provided hints or solutions to
selected ones. Complete Matlab solutions have been provided for the exercises that are marked as drills. The student's solution manual that contains solutions to the odd-numbered exercises is available separately as is the complete solution manual for the instructor
Electronic Text on a CD
This text-CD package includes a CD that contains the entire contents of the book neatly organized into an electronic text. In addition, the CD also contains concept demonstrations, Matlab drills, solutions, projects, and chapter review questions. The electronic material is supported by a well-designed graphical user interface that allows the user to navigate to any part of the text by clicking the mouse. The ability to read the text electronically, find any topic you need, do an on-line test, or perform a drill activity using Matlab makes this electronic text a useful instrument for learning linear algebra.
We believe that the use of technology in mathematics enriches the learning experience and encourages exploration of computationally hard problems that might not be easy to solve by hand. The experience of using technology to solve mathematical problems is an essential skill for today's graduates entering a high-tech dominated world. Although we have chosen Matlab as our technology tool, the printed text can also be used in a traditional setting. For the instructor who is interested in the use of technology in teaching, this CD contains a wealth of material for teaching linear algebra in a computer lab setting. It provides an interactive environment that encourages a hands-on approach.
Suggestions for Implementation
The book is suitable for a one-semester course in linear algebra. It may also be used for a one-quarter course by skipping certain sections in Chapters 5-10. We suggest the following guidelines for teaching the course in a computer lab setting. First, begin with demos in the electronic textbook to illustrate a new concept. Then advise the students to read related material in the text to reinforce the concept. Next, explain the Matlab operations needed in a given chapter. The list of basic Matlab operations is provided not only on the CD but also in the printed text. Pick a drill in a chapter and work it out fully in the class, and assign Matlab exercises to work in the lab or at home. Exercises marked as drills with a picture of a CD have Matlab solutions in the CD as well as at the end of the printed text. Solutions of drills using Mathematica/Maple are given on the Web page. Finally, encourage students to do the projects using Matlab.
S.K.Jain
A.D.Gunawardena
The purpose of this book is to provide an introduction to linear algebra, a branch of mathematics dealing with matrices and vector spaces. Matrices have been introduced here as a handy tool for solving systems of linear equations. But their utility goes far beyond this initial application. There is hardly any area of modem mathematics in which matrices do not have some application. They have also many applications in other disciplines, such as statistics, economics, engineering, physics, chemistry, biology, geology, and business.
The present text in linear algebra is designed for a general audience of sophomore-level students majoring in any area of art, science, or engineering. The only prerequisites are two or three years of high school mathematics with some knowledge of calculus.
A special feature of this book is that it can be used in a course taught in a traditional
manner as well as in a course using technology. Those using technology may refer to complete solutions of selected exercises (marked as drills) using Matlab at the end of the book, while others using Maple or Mathematica may refer to corresponding solutions on the web page. The readers would find the examples and solutions to drills using technology quite helpful and illustrative to solve similar problems. Concepts and practical methods for solving problems are illustrated through plenty of examples. The theorems and facts underlying these methods are clearly stated as they arise, but their proofs are provided in a separate section called "Proofs of Facts" near the end of thechapter.
Text Organization
The subject matter is laid out ill a leisurely manner with plenty of examples to illus-trate concepts and applications. Most of the sections contain a fairly large number of exercises, some of which relate to real-life problems. Chapters 1 and 2 deal with linear systems, leading naturally to matrices and to algebraic operations on matrices.
Chapter 3 introduces the vector space F" of n-tuples, linear dependence and indepen-dence of vectors. The concept of rank is introduced in Chapter 4 and is followed by more applications of elementary row operations in Chapter 5. Specifically, we show in Chapter 5 how to find the inverse of an invertible matrix, the LU-decomposition and full rank factorization of a matrix. Chapter 6 provides a working knowledge of determinants, which are later considered in a rigorous fashion in Chapter 10. Eigen-value problems and inner product spaces are given in Chapters 7 and 8, respectively.
An interesting feature of Chapter 7 is a method for finding eigenvalues without determinants. Two methods for finding the least-squares solution of an inconsistent linear system are given in Chapter 8. The first method is geometric and uses the notion of shortest distance; the second is algebraic and uses the concept of generalized in-verses. Vector spaces are revisited in Chapter 9, where a formal definition of a vector space is given and important examples of vector spaces of functions are considered.
In addition to the answers to all exercises, we have provided hints or solutions to
selected ones. Complete Matlab solutions have been provided for the exercises that are marked as drills. The student's solution manual that contains solutions to the odd-numbered exercises is available separately as is the complete solution manual for the instructor
Electronic Text on a CD
This text-CD package includes a CD that contains the entire contents of the book neatly organized into an electronic text. In addition, the CD also contains concept demonstrations, Matlab drills, solutions, projects, and chapter review questions. The electronic material is supported by a well-designed graphical user interface that allows the user to navigate to any part of the text by clicking the mouse. The ability to read the text electronically, find any topic you need, do an on-line test, or perform a drill activity using Matlab makes this electronic text a useful instrument for learning linear algebra.
We believe that the use of technology in mathematics enriches the learning experience and encourages exploration of computationally hard problems that might not be easy to solve by hand. The experience of using technology to solve mathematical problems is an essential skill for today's graduates entering a high-tech dominated world. Although we have chosen Matlab as our technology tool, the printed text can also be used in a traditional setting. For the instructor who is interested in the use of technology in teaching, this CD contains a wealth of material for teaching linear algebra in a computer lab setting. It provides an interactive environment that encourages a hands-on approach.
Suggestions for Implementation
The book is suitable for a one-semester course in linear algebra. It may also be used for a one-quarter course by skipping certain sections in Chapters 5-10. We suggest the following guidelines for teaching the course in a computer lab setting. First, begin with demos in the electronic textbook to illustrate a new concept. Then advise the students to read related material in the text to reinforce the concept. Next, explain the Matlab operations needed in a given chapter. The list of basic Matlab operations is provided not only on the CD but also in the printed text. Pick a drill in a chapter and work it out fully in the class, and assign Matlab exercises to work in the lab or at home. Exercises marked as drills with a picture of a CD have Matlab solutions in the CD as well as at the end of the printed text. Solutions of drills using Mathematica/Maple are given on the Web page. Finally, encourage students to do the projects using Matlab.
S.K.Jain
A.D.Gunawardena
