Introduction to linear algebra and differential equations 线性代数与微分方程导论

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Excellent introductory text for students with 1 year of calculus. Topics include complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions and boundary-value problems. Includes 48 black-and-white illustrations. Exercises with solutions. Index.

1 Complex Numbers
1.1 Introduction
1.2 The Algebra of Complex Numbers
1.3 The Geometry of Complex Numbers
1.4 Two-dimensional Vectors
1.5 Functions of a Complex Variable
1.6 Exponential Function
1.7 Power Series
2 Linear Algebraic Equations
2.1 Introduction
2.2 Matrices
2.3 Elimination Method
2.4 Determinants
2.5 Inverse of a Matrix
2.6 Existence and Uniqueness Theorems
3 Vector Spaces
3.1 Introduction
3.2 Three-dimensional Vectors
3.3 Axioms of a Vector Space
3.4 Dependence and Independence of Vectors
3.5 Basis and Dimension
3.6 Scalar Product
3.7 0rthonormal Bases
3.8 Infinite-dimensional Vector Spaces
4 Linear Transformations
4.1 Introduction
4.2 Definitions and Examples
4.3 Matrix Representations
4.4 Change of Bases
4.5 Characteristic Values and Characteristic Vectors
4.6 Symmetric and Hermitian Matrices
4.7 Jordan Forms
5 First Order Differential Equations
5.1 Introduction
5.2 An Example
5.3 Basic Definitions
5.4 First Order Linear Equations
5.5 First Order Nonlinear Equations
5.6 Applications of First Order Equations
5.7 Numerical Methods
5.8 Existence and Uniqueness
6 Linear Differential Equations
6.1 Introduction
6.2 General Theorems
6.3 Variation of Parameters
6.4 Equations with Constant Coefficients
6.5 Method of Undetermined Coefficients
6.6 Applications
6.7 Green's Funcuons
7 Laplace Transforms
8 Powe-Series Methods
9 Systems of Differential Equations
Answers and Hints for Selected Exercises
Index