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線性代數導引 版權信息
- ISBN：9787030721631
- 條形碼：9787030721631 ; 978-7-03-072163-1
- 裝幀：一般膠版紙
- 冊數：暫無
- 重量：暫無
- 所屬分類：>
線性代數導引 內容簡介
近年來，隨著能源環境問題日益凸顯和輕量化設計制造的需求日益迫切，航空航天、軌道交通、節能汽車等高技術領域對原位鋁基復合材料的需求潛力巨大，且對其綜合性能的要求也越來越高。本書較系統、詳細地介紹了原位鋁基復合材料的體系設計、材料開發、制備技術、凝固組織、塑變加工及性能。全書共八章，主要內容包括：原位反應體系的設計與開發、電磁法合成原位鋁基復合材料、高能超聲法合成原位鋁基復合材料、聲磁耦合法合成原位鋁基復合材料、原位鋁基復合材料的凝固組織及界面結構、塑變加工對原位鋁基復合材料組織的影響、原位鋁基復合材料的力學性能、原位鋁基復合材料的磨損性能。內容豐富、新穎，具有系統性和前瞻性，反映了作者團隊二十余年來在原位鋁基復合材料領域的科研成果。
線性代數導引 目錄
Contents
Chapter 1 Linear Systems and Matrices 1
1.1 Introduction to Linear Systems and Matrices 1
1.1.1 Linear equations and linear systems 1
1.1.2 Matrices 3
1.1.3 Elementary row operations 4
1.2 Gauss-Jordan Elimination 5
1.2.1 Reduced row-echelon form 5
1.2.2 Gauss-Jordan elimination 6
1.2.3 Homogeneous linear systems 9
1.3 Matrix Operations 11
1.3.1 Operations on matrices 11
1.3.2 Partition of matrices 13
1.3.3 Matrix product by columns and by rows 13
1.3.4 Matrix product of partitioned matrices 14
1.3.5 Matrix form of a linear system 15
1.3.6 Transpose and trace of a matrix 16
1.4 Rules of Matrix Operations and Inverses 18
1.4.1 Basic properties of matrix operations 19
1.4.2 Identity matrix and zero matrix 20
1.4.3 Inverse of a matrix 21
1.4.4 Powers of a matrix 23
1.5 Elementary Matrices and a Method for Finding A.1 24
1.5.1 Elementary matrices and their properties 24
1.5.2 Main theorem of invertibility 26
1.5.3 A method for finding A.1 27
1.6 Further Results on Systems and Invertibility 28
1.6.1 A basic theorem 28
1.6.2 Properties of invertible matrices 29
1.7 Some Special Matrices 31
1.7.1 Diagonal and triangular matrices 32
1.7.2 Symmetric matrix 34
Exercises 35
Chapter 2 Determinants 42
2.1 Determinant Function 42
2.1.1 Permutation， inversion， and elementary product 42
2.1.2 Definition of determinant function 44
2.2 Evaluation of Determinants 44
2.2.1 Elementary theorems 44
2.2.2 A method for evaluating determinants 46
2.3 Properties of Determinants 46
2.3.1 Basic properties 47
2.3.2 Determinant of a matrix product 48
2.3.3 Summary 50
2.4 Cofactor Expansions and Cramer’s Rule 51
2.4.1 Cofactors 51
2.4.2 Cofactor expansions 51
2.4.3 Adjoint of a matrix 53
2.4.4 Cramer’s rule 54
Exercises 55
Chapter 3 Euclidean Vector Spaces 61
3.1 Euclidean n-Space 61
3.1.1 n-vector space 61
3.1.2 Euclidean n-space 62
3.1.3 Norm， distance， angle， and orthogonality 63
3.1.4 Some remarks 65
3.2 Linear Transformations from Rn to Rm 66
3.2.1 Linear transformations from Rn to Rm 66
3.2.2 Some important linear transformations 67
3.2.3 Compositions of linear transformations 69
3.3 Properties of Transformations 70
3.3.1 Linearity conditions 70
3.3.2 Example 71
3.3.3 One-to-one transformations 72
3.3.4 Summary 73
Exercises 74
Chapter 4 General Vector Spaces 79
4.1 Real Vector Spaces 79
4.1.1 Vector space axioms 79
4.1.2 Some properties 81
4.2 Subspaces 81
4.2.1 Definition of subspace 82
4.2.2 Linear combinations 83
4.3 Linear Independence 85
4.3.1 Linear independence and linear dependence 86
4.3.2 Some theorems 87
4.4 Basis and Dimension 88
4.4.1 Basis for vector space 88
4.4.2 Coordinates 89
4.4.3 Dimension 91
4.4.4 Some fundamental theorems 93
4.4.5 Dimension theorem for subspaces 95
4.5 Row Space， Column Space， and Nullspace 97
4.5.1 Definition of row space， column space， and nullspace 97
4.5.2 Relation between solutions of Ax = 0 and Ax=b 98
4.5.3 Bases for three spaces 100
4.5.4 A procedure for finding a basis for span(S) 102
4.6 Rank and Nullity 103
4.6.1 Rank and nullity 104
4.6.2 Rank for matrix operations 106
4.6.3 Consistency theorems 107
4.6.4 Summary 109
Exercises 110
Chapter 5 Inner Product Spaces 115
5.1 Inner Products 115
5.1.1 General inner products 115
5.1.2 Examples 116
5.2 Angle and Orthogonality 119
5.2.1 Angle between two vectors and orthogonality 119
5.2.2 Properties of length， distance， and orthogonality 120
5.2.3 Complement 121
5.3 Orthogonal Bases and Gram-Schmidt Process 122
5.3.1 Orthogonal and orthonormal bases 122
5.3.2 Projection theorem 125
5.3.3 Gram-Schmidt process 128
5.3.4 QR-decomposition 130
5.4 Best Approximation and Least Squares 133
5.4.1 Orthogonal projections viewed as approximations 134
5.4.2 Least squares solutions of linear systems 135
5.4.3 Uniqueness of least squares solutions 136
5.5 Orthogonal Matrices and Change of Basis. 138
5.5.1 Orthogonal matrices 138
5.5.2 Change of basis 140
Exercises 144
Chapter 6 Eigenvalues and Eigenvectors 149
6.1 Eigenvalues and Eigenvectors 149
6.1.1 Introduction to eigenvalues and eigenvectors 149
6.1.2 Two theorems concerned with eigenvalues 150
6.1.3 Bases for eigenspaces 151
6.2 Diagonalization 152
6.2.1 Diagonalization problem 152
6.2.2 Procedure for diagonalization 153
6.2.3 Two theorems concerned with diagonalization 155
6.3 Orthogonal Diagonalization 156
6.4 Jordan Decomposition Theorem 160
Exercises 162
Chapter 7 Linear Transformations 166
7.1 General Linear Transformations 166
7.1.1 Introduction to linear transformations 166
7.1.
Chapter 1 Linear Systems and Matrices 1
1.1 Introduction to Linear Systems and Matrices 1
1.1.1 Linear equations and linear systems 1
1.1.2 Matrices 3
1.1.3 Elementary row operations 4
1.2 Gauss-Jordan Elimination 5
1.2.1 Reduced row-echelon form 5
1.2.2 Gauss-Jordan elimination 6
1.2.3 Homogeneous linear systems 9
1.3 Matrix Operations 11
1.3.1 Operations on matrices 11
1.3.2 Partition of matrices 13
1.3.3 Matrix product by columns and by rows 13
1.3.4 Matrix product of partitioned matrices 14
1.3.5 Matrix form of a linear system 15
1.3.6 Transpose and trace of a matrix 16
1.4 Rules of Matrix Operations and Inverses 18
1.4.1 Basic properties of matrix operations 19
1.4.2 Identity matrix and zero matrix 20
1.4.3 Inverse of a matrix 21
1.4.4 Powers of a matrix 23
1.5 Elementary Matrices and a Method for Finding A.1 24
1.5.1 Elementary matrices and their properties 24
1.5.2 Main theorem of invertibility 26
1.5.3 A method for finding A.1 27
1.6 Further Results on Systems and Invertibility 28
1.6.1 A basic theorem 28
1.6.2 Properties of invertible matrices 29
1.7 Some Special Matrices 31
1.7.1 Diagonal and triangular matrices 32
1.7.2 Symmetric matrix 34
Exercises 35
Chapter 2 Determinants 42
2.1 Determinant Function 42
2.1.1 Permutation， inversion， and elementary product 42
2.1.2 Definition of determinant function 44
2.2 Evaluation of Determinants 44
2.2.1 Elementary theorems 44
2.2.2 A method for evaluating determinants 46
2.3 Properties of Determinants 46
2.3.1 Basic properties 47
2.3.2 Determinant of a matrix product 48
2.3.3 Summary 50
2.4 Cofactor Expansions and Cramer’s Rule 51
2.4.1 Cofactors 51
2.4.2 Cofactor expansions 51
2.4.3 Adjoint of a matrix 53
2.4.4 Cramer’s rule 54
Exercises 55
Chapter 3 Euclidean Vector Spaces 61
3.1 Euclidean n-Space 61
3.1.1 n-vector space 61
3.1.2 Euclidean n-space 62
3.1.3 Norm， distance， angle， and orthogonality 63
3.1.4 Some remarks 65
3.2 Linear Transformations from Rn to Rm 66
3.2.1 Linear transformations from Rn to Rm 66
3.2.2 Some important linear transformations 67
3.2.3 Compositions of linear transformations 69
3.3 Properties of Transformations 70
3.3.1 Linearity conditions 70
3.3.2 Example 71
3.3.3 One-to-one transformations 72
3.3.4 Summary 73
Exercises 74
Chapter 4 General Vector Spaces 79
4.1 Real Vector Spaces 79
4.1.1 Vector space axioms 79
4.1.2 Some properties 81
4.2 Subspaces 81
4.2.1 Definition of subspace 82
4.2.2 Linear combinations 83
4.3 Linear Independence 85
4.3.1 Linear independence and linear dependence 86
4.3.2 Some theorems 87
4.4 Basis and Dimension 88
4.4.1 Basis for vector space 88
4.4.2 Coordinates 89
4.4.3 Dimension 91
4.4.4 Some fundamental theorems 93
4.4.5 Dimension theorem for subspaces 95
4.5 Row Space， Column Space， and Nullspace 97
4.5.1 Definition of row space， column space， and nullspace 97
4.5.2 Relation between solutions of Ax = 0 and Ax=b 98
4.5.3 Bases for three spaces 100
4.5.4 A procedure for finding a basis for span(S) 102
4.6 Rank and Nullity 103
4.6.1 Rank and nullity 104
4.6.2 Rank for matrix operations 106
4.6.3 Consistency theorems 107
4.6.4 Summary 109
Exercises 110
Chapter 5 Inner Product Spaces 115
5.1 Inner Products 115
5.1.1 General inner products 115
5.1.2 Examples 116
5.2 Angle and Orthogonality 119
5.2.1 Angle between two vectors and orthogonality 119
5.2.2 Properties of length， distance， and orthogonality 120
5.2.3 Complement 121
5.3 Orthogonal Bases and Gram-Schmidt Process 122
5.3.1 Orthogonal and orthonormal bases 122
5.3.2 Projection theorem 125
5.3.3 Gram-Schmidt process 128
5.3.4 QR-decomposition 130
5.4 Best Approximation and Least Squares 133
5.4.1 Orthogonal projections viewed as approximations 134
5.4.2 Least squares solutions of linear systems 135
5.4.3 Uniqueness of least squares solutions 136
5.5 Orthogonal Matrices and Change of Basis. 138
5.5.1 Orthogonal matrices 138
5.5.2 Change of basis 140
Exercises 144
Chapter 6 Eigenvalues and Eigenvectors 149
6.1 Eigenvalues and Eigenvectors 149
6.1.1 Introduction to eigenvalues and eigenvectors 149
6.1.2 Two theorems concerned with eigenvalues 150
6.1.3 Bases for eigenspaces 151
6.2 Diagonalization 152
6.2.1 Diagonalization problem 152
6.2.2 Procedure for diagonalization 153
6.2.3 Two theorems concerned with diagonalization 155
6.3 Orthogonal Diagonalization 156
6.4 Jordan Decomposition Theorem 160
Exercises 162
Chapter 7 Linear Transformations 166
7.1 General Linear Transformations 166
7.1.1 Introduction to linear transformations 166
7.1.
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線性代數導引 節選
Chapter 1 Linear Systems and Matrices “No beginner’s course in mathematics can do without linear algebra，” —Lars Garding “Matrices act They don’t just sit there.” —Gilbert Strang Solving linear systems (a system of linear equations) is the most important problem of linear algebra and possibly of applied mathematics as well. Usually， information in a linear system is often arranged into a rectangular array， called a “matrix”. The matrix is particularly important in developing computer programs to solve linear systems with huge sizes because computers are suitable to manage numerical data in arrays. Moreover， matrices are not only a simple tool for solving linear systems but also mathematical objects in their own right. In fact， matrix theory has a variety of applications in science， engineering， and mathematics. Therefore， we begin our study on linear systems and matrices in the first chapter. 1.1 Introduction to Linear Systems and Matrices Let IR denote the set of real numbers. We now introduce linear equations， linear systems， and matrices. 1.1.1 Linear equations and linear systems We consider where are coefficients，are variables (unknowns)， n is a positive integer， and 6 G R is a constant. An equation of this form is called a Zinear equation， in which all variables occur to the first power.， the linear equation is called a homogeneous linear equation. A sequence of numbers si， sn is called a solution of the equation if，xn = sn such that The set of all solutions of the equation is called the solution set of the equation. In the book， we always use example(s) to make our points clear. Example We consider the following linear equations: (a) (b) It is easy to see that the solution set of (a) is a line in xy-plane and the solution set of (b) is a plane in xyz-space. We next consider the following m linear equations in n variables: (1-1) where are coefficients，are variables， and bi are constants. A system of linear equations in this form is called a linear system. A sequence of numbers si，is called a solution of the system if，is a solution of each equation in the system. A linear system is said to be consistent if it has at least one solution.Otherwise， a linear system is said to be inconsistent if it has no solution. Example Consider the following linear system The graphs of these equations are lines called li and We have three possible cases of lines l\ and I2 in xy-plane. See Figure 1.1. When l\ and I2 are parallel， there is no solution of the system. When li and I2 intersect at only one point， there is exactly one solution of the system. When l1 and I2 coincide， there are infinitely many solutions of the system. Figure 1.1 1.1.2 Matrices The term matrix was first introduced by a British mathematician James Sylvester in the 19th century. Another British mathematician Arthur Cayley developed basic algebraic operations on matrices in the 1850s. Up to now， matrices have become the language to know. Definition A matrix is a rectangular array of numbers. The numbers in the array are called the entries in the matrix. Remark The size of a matrix is described in terms of the number of rows and columns it contains. Usually， a matrix with m rows and n columns is called an m x n matrix. If A is an m x n matrix， then we denote the entry in row i and column j of A by the symbol (A)ij = a々.Moreover， a matrix with real entries will be called a real matrix and the set of all m x n real matrices will be denoted by the symbol Rmxn. For instance， a matrix A in IRmxn can be written as where G IR for any i and j. When compactness of notation is desired， the preceding matrix can be written as We now introduce some important matrices with special sizes. A row matrix is a general 1 x n matrix a given by The main diagonal of the square matrix A is the set of entries an， (1.2) For linear system (1.1)， we can write it briefly as the following matrix form which is called the augmented matrix of (1.1). Remark When we construct an augmented matrix associated with a given linear system， the unknowns must be written in the same order in each equation and the constants must be on the right. 1.1.3 Elementary row operations In order to solve a linear system efficiently， we replace the given system with its augmented matrix and then solve the same system by operating on the rows of the augmented matrix. There are three elementary row operations on matrices defined as follows: (1) Interchange two rows. (2) Multiply a row by a nonzero number. (3) Add a multiple of one row to another row. By using elementary row operations， we can always reduce the augmented matrix of a given system to a simpler augmented matrix from which the solution of the system is evident. See the following example. 1.2 Gauss-Jordan Elimination In this s
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