基本介紹
- 書名：Linear Algebra
- 出版社：高等教育出版社
- 頁數：355頁
- 開本：16
- 品牌：高等教育出版社
- 作者：彭國華 李德琅
- 出版日期：2006年5月1日
- 語種：簡體中文, 英語
- ISBN：7040192829
- 定價：17.90
內容簡介,圖書目錄,文摘,序言,編輯推薦,目錄,
內容簡介
《Linear Algebra》為普通高等教育十一五國家級規劃教材之一。
圖書目錄
1 Integers and Polynomials
1.1 Integers
1.2 N'umber Fields
1.3 Polynomial
1.4 Polynomial Functions and Roots
1.5 Polvnomials over Rational Number Field
1.6 Polynomials of Several Variables
1.7 Symmetric Polynomials
1.8 Exercises
2 Systems of Linear Equations
2.1 Systems of Linear Equations and Elimination
2.2 Vectors
2.3 Matrices
2.4 Structure of Solutions of A System of Linear Equations
2.5 Exercises
3 Linear Maps, Matrices and Determinants
3. 1 Linear Maps of Vector Spaces and Matrices
3.2 Operations of Linear Maps and Matrices
3.3 Partitioned Matrices
3, 4 Elementary Matrices and Invertible Matrices
3.5 Determinants
3.5.1 Permutation and Determinant
3.5.2 Properties of Determinant
3.5.3 Expansion of Determinant
3.5.4 Applications of Determinant
3.6 Exercises
4 Linear Spaces and Linear Maps
4. 1 Linear Spaces
4.2 Dimension, Basis, Coordinates
4.3 Basis Change and Coordinate Transformations
4.4 Linear Maps and Isomorphism
4.5 Matrices of Linear Maps
4.6 Subspaces and Direct Sum
4.7 Space Decomposition and Partitioned Matrices
4.8 Exercises
5 Linear Transformations
5.1 Linear Transformations
5.2 Similarity of Matrices
5.3 ),-Matrices
5.4 Eigenvalues, Eigenvectors and Characteristic Polynomials
5.5 Invariant Subspaces
5.6 Equivalence of λ-matrices
5.7 Invariant Factors and Elementary Divisors
5.8 Condition for Similarity of Matrices
5.9 Jordan Canonical Forms of Matrices
5.10 Rat iona! Canonical Forms of Matrices
5.11 Exercises
6 Euclidean Spaces
6.1 Inner Product and Basic Properties
6.2 Orthogonal Bases and Schmidt Orthonormalization
6.3 Subspaces and Orthogonal Complements
6.4 Isometry and Orthogonal Transformations
6.5 Symmetric Matrices and Symmetric Transformations
6.6 The Method of Least Squares——System of Linear Equations
Revisited
6.7 A Brief Introduction to Unitary Spaces
6.8 Exercises
7 Linear Forms, Bilinear Forms and Quadratic Forms
7.1 Linear Forms and the Dual Space
7.2 Bilinear Forms
7.3 Symmetric Bilinear Forms
7.4 Quadratic Forms
7.5 Quadratic Forms over Real and Complex Number Fields
7.6 Positive Definite Quadratic Forms over Real Number Field
7.7 Exercises
Bibliography
Index
1.1 Integers
1.2 N'umber Fields
1.3 Polynomial
1.4 Polynomial Functions and Roots
1.5 Polvnomials over Rational Number Field
1.6 Polynomials of Several Variables
1.7 Symmetric Polynomials
1.8 Exercises
2 Systems of Linear Equations
2.1 Systems of Linear Equations and Elimination
2.2 Vectors
2.3 Matrices
2.4 Structure of Solutions of A System of Linear Equations
2.5 Exercises
3 Linear Maps, Matrices and Determinants
3. 1 Linear Maps of Vector Spaces and Matrices
3.2 Operations of Linear Maps and Matrices
3.3 Partitioned Matrices
3, 4 Elementary Matrices and Invertible Matrices
3.5 Determinants
3.5.1 Permutation and Determinant
3.5.2 Properties of Determinant
3.5.3 Expansion of Determinant
3.5.4 Applications of Determinant
3.6 Exercises
4 Linear Spaces and Linear Maps
4. 1 Linear Spaces
4.2 Dimension, Basis, Coordinates
4.3 Basis Change and Coordinate Transformations
4.4 Linear Maps and Isomorphism
4.5 Matrices of Linear Maps
4.6 Subspaces and Direct Sum
4.7 Space Decomposition and Partitioned Matrices
4.8 Exercises
5 Linear Transformations
5.1 Linear Transformations
5.2 Similarity of Matrices
5.3 ),-Matrices
5.4 Eigenvalues, Eigenvectors and Characteristic Polynomials
5.5 Invariant Subspaces
5.6 Equivalence of λ-matrices
5.7 Invariant Factors and Elementary Divisors
5.8 Condition for Similarity of Matrices
5.9 Jordan Canonical Forms of Matrices
5.10 Rat iona! Canonical Forms of Matrices
5.11 Exercises
6 Euclidean Spaces
6.1 Inner Product and Basic Properties
6.2 Orthogonal Bases and Schmidt Orthonormalization
6.3 Subspaces and Orthogonal Complements
6.4 Isometry and Orthogonal Transformations
6.5 Symmetric Matrices and Symmetric Transformations
6.6 The Method of Least Squares——System of Linear Equations
Revisited
6.7 A Brief Introduction to Unitary Spaces
6.8 Exercises
7 Linear Forms, Bilinear Forms and Quadratic Forms
7.1 Linear Forms and the Dual Space
7.2 Bilinear Forms
7.3 Symmetric Bilinear Forms
7.4 Quadratic Forms
7.5 Quadratic Forms over Real and Complex Number Fields
7.6 Positive Definite Quadratic Forms over Real Number Field
7.7 Exercises
Bibliography
Index
文摘
插圖：
84．Use the least squares method to find the best choice of a line Y=a。+a1xto fit the（X，Y）一data points（-1，1），（0，0），（1，3），（2，4）．Plot the 1ineand the data points in the（x，y）一plane．
85．The owner of a rapidly expanding business finds that for the first fivemonths of the year his sales are RMB 4000，4400，5200，6400 and8000．He plots these figures on a graph and conjectures that for the restof the year his sales curve can be approximated by a quadratic polynomia1．Find the least squares quadratic polynomial fit to the sales curve anduse it to project the sales for the 12th month of the year．
86．When the space shuttle Challenger exploded in 1986，one of the criticisms made of NASA’S decision to Launch was in the way the analysis ofnumber of Oring failures versus temperature was made（of course，Oring failure caused the explosion）．Four Oring failures will cause therocket tO explode．NASA had data from 24 previous flights．
84．Use the least squares method to find the best choice of a line Y=a。+a1xto fit the（X，Y）一data points（-1，1），（0，0），（1，3），（2，4）．Plot the 1ineand the data points in the（x，y）一plane．
85．The owner of a rapidly expanding business finds that for the first fivemonths of the year his sales are RMB 4000，4400，5200，6400 and8000．He plots these figures on a graph and conjectures that for the restof the year his sales curve can be approximated by a quadratic polynomia1．Find the least squares quadratic polynomial fit to the sales curve anduse it to project the sales for the 12th month of the year．
86．When the space shuttle Challenger exploded in 1986，one of the criticisms made of NASA’S decision to Launch was in the way the analysis ofnumber of Oring failures versus temperature was made（of course，Oring failure caused the explosion）．Four Oring failures will cause therocket tO explode．NASA had data from 24 previous flights．
序言
高等代數是數學系本科一年級的課程。在教學過程中我們逐漸感到有必要編寫一本新的高等代數教材。一方面是為了與中學教學內容的銜接，另一方面也是為了順應時代的需要。由於對課程內容和教學理念有許多一致的看法，兩位作者準備聯合編寫一本高等代數教材。講義初稿從2000年起著手編寫，2001年9月完成，並於當年秋季起一直在四川大學數學學院一年級所有班級中連續使用。由於書稿是在電腦上直接寫成的，而敲打漢字對我們來說又是件痛苦的事，因此我們就順便用英文寫成。從近五年的連續使用情況來看，學生反映教材章節安排自然合理，邏輯清晰，舉例適當。大多數學生認為習題新穎、量大，難易結合且和教學內容互補搭配。
我們以如何求解線性方程組為出發點，進而考慮解的結構，自然引申出向量、矩陣、行列式、線性空間等概念並展開討論。第一章預先討論了一元多項式和多元多項式的基本概念和結論，為後面學習線性代數做準備。在第二章里，我們以討論解線性方程組的解為中心，引入了向量和矩陣的概念並討論了它們的基本關係和性質。第三章主要討論了矩陣的運算和行列式。我們首先建立了在固定的標準單位向量組下向量空間上的線性映射和矩陣的一一對應關係，然後將矩陣的加法和乘積自然定義為線性映射加和乘的矩陣。進而藉助映射誘導出矩陣的許多性質。這一章里我們還專門討論了分塊矩陣的運算準則和例子。在討論行列式時，我們藉助初等矩陣，給出矩陣乘積行列式的公式，並沒有用到行列式的定義。這些是與大多數的其他同類教材不同的地方。第四章、第五章分別討論了線性空間、線性映射和線性變換。在第五章里我們還討論了矩陣、若爾當標準形和一般數域上的有理標準形。第六章是關於歐幾里得空間的，主要包含內積空間、正交變換、對稱變換、正交矩陣、對稱矩陣等基本內容。我們把二次型放到了第七章。這一章先講了線性型和雙線性型。作為套用，我們討論了二次型和正定二次型的基本性質，包括二次型的標準形問題。
我們以如何求解線性方程組為出發點，進而考慮解的結構，自然引申出向量、矩陣、行列式、線性空間等概念並展開討論。第一章預先討論了一元多項式和多元多項式的基本概念和結論，為後面學習線性代數做準備。在第二章里，我們以討論解線性方程組的解為中心，引入了向量和矩陣的概念並討論了它們的基本關係和性質。第三章主要討論了矩陣的運算和行列式。我們首先建立了在固定的標準單位向量組下向量空間上的線性映射和矩陣的一一對應關係，然後將矩陣的加法和乘積自然定義為線性映射加和乘的矩陣。進而藉助映射誘導出矩陣的許多性質。這一章里我們還專門討論了分塊矩陣的運算準則和例子。在討論行列式時，我們藉助初等矩陣，給出矩陣乘積行列式的公式，並沒有用到行列式的定義。這些是與大多數的其他同類教材不同的地方。第四章、第五章分別討論了線性空間、線性映射和線性變換。在第五章里我們還討論了矩陣、若爾當標準形和一般數域上的有理標準形。第六章是關於歐幾里得空間的，主要包含內積空間、正交變換、對稱變換、正交矩陣、對稱矩陣等基本內容。我們把二次型放到了第七章。這一章先講了線性型和雙線性型。作為套用，我們討論了二次型和正定二次型的基本性質，包括二次型的標準形問題。
編輯推薦
《Linear Algebra》是由高等學校出版社出版的。
目錄
Chapter 1 System of Linear Equations and Elimination Method
1.1 Solving System of Linear Equations with Elimination Method
1.1.1 Linear System with Two Unknowns
1.1.2 Gauss-Jordan Elimination Method
1.2 Applications
Practice 1
Chapter 2 Matrices
2.1 Basic Concepts
2.1.1 Matrices
2.1.2 Special Matrices
2.1.3 Problems Related to Matrices
2.2 Basic Operations
2.2.1 Definitions
2.2.2 Rules of Operations
2.2.3 Applications
2.3 Matrix Inverses
2.3.1 lnvertible Matrices
2.3.2 Orthogonal Matrices
2.4 Blocks and Sub-matrices
2.4.1 Block Operations
2.4.2 Column Blocks
2.4.3 Sub-matrices
2.5 Elementary Operations and Elementary Matrices
2.5.1 Definitions and Properties
2.5.2 Equivalent Normal Form for Matrices
2.5.3 Invertible Matrices Revisit
2.5.4 Unique solution for n x n linear systems
2.6 Applications(Input - output Analysis)
Practice 2
Chapter 3 Determinants
3.1 Definitions and Properties of Determinants
3.1.1 Definitions
3.1.2 Propertie
3.2 Evaluation of Determinants
3.3 Applications
3.3.1 Adjugate Matrices and Inverse Formula
3.3.2 Cramer's Rule
3.3.3 Summary
Practice 3
Chapter 4 Rank of a Matrix and Solutions for Linear Systems
4.1 Rank of a Matrix
4.1.1 Concepts
4.1.2 Computations
4.2 Solutions of Linear Systems
4.2.1 Homogeneous Systems
4.2.2 Non-homogeneous Systems
Practice 4
Chapter 5 Vector Spaces
5.1 Concepts
5.2 Linear Dependence and Linear Independence
5.2.1 Concepts
5.2.2 Properties
5.2.3 Rank of a Set of Vectors
5.2.4 Row and Column Ranks of a Matrix
5.3 Bases and Dimensions of Vector Spaces
5.3.1 Bases and Dimensions
5.3.2 Revisit Solutions for Linear Systems
5.4 Inner Products
5.4.1 Review
5.4.2 Inner Products and Orthogonal Matrices
5.4.3 Four Basic Subspaces
Practice 5
Chapter 6 Eigenvalues
6.1 Eigenvalues and Eigenvectors
6.2 Diagonalizations'
6.2.1 Similar Matrices and Diagonal Forms
6.2.2 Applications
6.3 Real Symmetric Matrices and Quadratic Forms
6.3..1 Canonical Forms for Real Symmetric Matrices
6.3.2 Quadratic Forms
6.3.3 Quadratic Expressions and Their Canonical Forms
6.4 Positive Definite Matrices and Classification of Quadratic Forms
6.4.1 Positive Definite Matrices
6.4.2 Optimization
6.4.3 Generalized Eigenvalue Problems
Practice 6
Chapter 7 Linear Transformations
7.1 Basic Concepts of Linear Transformations
7.1.1 Linear Transformations
7.1.2 Range and Kernel for a Linear Transformation
7.2 Linear Transformations and Matrices
7.2.1 Coordinate Vectors
7.2.2 The Matrix Representations for Linear Transformations
7.2.3 Engenvalues and Eigenvectors of a Linear Transformation
Practice 7
References
1.1 Solving System of Linear Equations with Elimination Method
1.1.1 Linear System with Two Unknowns
1.1.2 Gauss-Jordan Elimination Method
1.2 Applications
Practice 1
Chapter 2 Matrices
2.1 Basic Concepts
2.1.1 Matrices
2.1.2 Special Matrices
2.1.3 Problems Related to Matrices
2.2 Basic Operations
2.2.1 Definitions
2.2.2 Rules of Operations
2.2.3 Applications
2.3 Matrix Inverses
2.3.1 lnvertible Matrices
2.3.2 Orthogonal Matrices
2.4 Blocks and Sub-matrices
2.4.1 Block Operations
2.4.2 Column Blocks
2.4.3 Sub-matrices
2.5 Elementary Operations and Elementary Matrices
2.5.1 Definitions and Properties
2.5.2 Equivalent Normal Form for Matrices
2.5.3 Invertible Matrices Revisit
2.5.4 Unique solution for n x n linear systems
2.6 Applications(Input - output Analysis)
Practice 2
Chapter 3 Determinants
3.1 Definitions and Properties of Determinants
3.1.1 Definitions
3.1.2 Propertie
3.2 Evaluation of Determinants
3.3 Applications
3.3.1 Adjugate Matrices and Inverse Formula
3.3.2 Cramer's Rule
3.3.3 Summary
Practice 3
Chapter 4 Rank of a Matrix and Solutions for Linear Systems
4.1 Rank of a Matrix
4.1.1 Concepts
4.1.2 Computations
4.2 Solutions of Linear Systems
4.2.1 Homogeneous Systems
4.2.2 Non-homogeneous Systems
Practice 4
Chapter 5 Vector Spaces
5.1 Concepts
5.2 Linear Dependence and Linear Independence
5.2.1 Concepts
5.2.2 Properties
5.2.3 Rank of a Set of Vectors
5.2.4 Row and Column Ranks of a Matrix
5.3 Bases and Dimensions of Vector Spaces
5.3.1 Bases and Dimensions
5.3.2 Revisit Solutions for Linear Systems
5.4 Inner Products
5.4.1 Review
5.4.2 Inner Products and Orthogonal Matrices
5.4.3 Four Basic Subspaces
Practice 5
Chapter 6 Eigenvalues
6.1 Eigenvalues and Eigenvectors
6.2 Diagonalizations'
6.2.1 Similar Matrices and Diagonal Forms
6.2.2 Applications
6.3 Real Symmetric Matrices and Quadratic Forms
6.3..1 Canonical Forms for Real Symmetric Matrices
6.3.2 Quadratic Forms
6.3.3 Quadratic Expressions and Their Canonical Forms
6.4 Positive Definite Matrices and Classification of Quadratic Forms
6.4.1 Positive Definite Matrices
6.4.2 Optimization
6.4.3 Generalized Eigenvalue Problems
Practice 6
Chapter 7 Linear Transformations
7.1 Basic Concepts of Linear Transformations
7.1.1 Linear Transformations
7.1.2 Range and Kernel for a Linear Transformation
7.2 Linear Transformations and Matrices
7.2.1 Coordinate Vectors
7.2.2 The Matrix Representations for Linear Transformations
7.2.3 Engenvalues and Eigenvectors of a Linear Transformation
Practice 7
References
