Preface......Page 6
Acknowledgements......Page 8
Contents......Page 9
Notations and Abbreviations......Page 12
1.1 Introduction to Set Theory......Page 14
1.1.1 Set Operations and Other Properties......Page 15
1.2 Equivalence Relations......Page 18
1.3 Functions or Maps......Page 21
1.3.1 Classification of Functions or Maps......Page 22
1.4 Well-Ordering Principle and Mathematical Induction......Page 26
References......Page 29
2.1 Basic Definitions......Page 31
2.2 Subgroups......Page 33
2.3 Homomorphisms......Page 38
2.4 The Isomorphism Theorems......Page 41
References......Page 42
3.1 Basic Definitions......Page 43
3.2 Ideals, Homomorphisms and Rings......Page 45
3.3 Isomorphism Theorems in Rings......Page 47
3.4.1 Divisibility......Page 48
3.4.3 Greatest Common Divisor......Page 50
3.4.4 Least Common Multiple......Page 54
3.5 Polynomials Rings......Page 56
References......Page 59
4.1 Properties of Algebraic Operations with Real Numbers......Page 60
4.2 The Set mathbbRn and Linear Operations......Page 61
4.2.1 Linear Operations in mathbbRn......Page 62
4.3 Background of Matrix Operations......Page 64
4.4 Gauss-Jordan Method......Page 70
4.5 Definitions......Page 74
References......Page 75
5.1 Permutations Group......Page 76
5.2 Determinants......Page 78
References......Page 84
6.1 Vector Spaces and Subspaces......Page 85
6.2 Generated Subspace......Page 89
6.3 Linear Dependence and Independence......Page 90
6.4 Bases and Dimension......Page 91
6.5 Quotient Space......Page 95
6.6 Cayley-Hamilton Theorem......Page 97
6.7 Euclidean Spaces......Page 98
6.8 GramSchmidt Process......Page 100
References......Page 101
7.1 Background......Page 102
7.2 Kernel and Image......Page 103
7.3 Linear Operators......Page 109
7.4 Associate Matrix......Page 111
References......Page 115
8.1 Matrix Diagonalization......Page 117
8.2 Jordan Canonical Form......Page 124
8.2.1 Generalized Eigenvectors......Page 126
8.2.2 Dot Diagram Method......Page 128
References......Page 131
9.1.1 Free Fall......Page 132
9.1.2 Simple Pendulum Problem......Page 133
9.1.3 Friction Problem......Page 134
9.2 Definitions......Page 135
9.3 Separable Differential Equations......Page 136
9.4 Homogeneous Equations......Page 137
9.5 Exact Equations......Page 139
9.6 Linear Differential Equations......Page 141
9.7 Homogeneous Second Order Linear Differential Equations......Page 144
9.8 Variation of Parameters Method......Page 146
9.9 Initial Value Problem......Page 148
9.10 Indeterminate Coefficients......Page 152
9.11.1 Some Criterions of Convergence of Series......Page 153
9.11.2 Solution of First and Second Order Differential Equations......Page 156
9.12 Picard's Method......Page 161
9.13 Convergence of Picard's Iterations......Page 163
References......Page 168
10.1 Algebraic and Transcendental Field Extensions......Page 169
10.2 Basic Notions of Differential Algebra......Page 171
10.3 Definitions and Properties of Single-Input Single-Output Systems......Page 173
10.5 Invertible Systems......Page 174
10.6 Realization......Page 178
10.7 Generalized Controller and Observer Canonical Forms......Page 179
10.8 Linearization by Dynamical Feedback of Nonlinear Dynamics......Page 180
References......Page 188
A.1 Summation......Page 189
A.2 Kronecker Delta......Page 193
A.3 Matrices......Page 195
Index......Page 198