Nguồn ngoài
Applied Linear Algebra and Differential Equations
These lecture notes are for a mathematics course offered to second-year engineering students. Material from our usual courses on linear algebra and differential equations have been combined into a single course (essentially, two half-semester courses) at the request of our Engineering School. I have tried my best to select the most essential and interesting topics from both courses, and to show how knowledge of linear algebra can improve students’ understanding of differential equations.
- Tác giả
- Jeffrey R. Chasnov
- Nhà xuất bản
- Hong Kong University of Science and Technology
- Nguồn
- LibreTexts
- Giấy phép
- CC BY
- Ngôn ngữ
- EN
- Danh mục
- Khoa học cơ bản
Mục lục· 67 mục
- 02: I. A Short Mathematical Review
- 02: II. Linear Algebra
- 01: Matrices
- 1.01: Definition of a Matrix
- 1.02: Addition and Multiplication of Matrices
- 1.03: General Notation Transposes and Inverses
- 1.03: The Identity Matrix and the Zero Matrix
- 1.04: Rotation Matrices and Orthogonal Matrices
- 1.06: Matrix Representation of Complex Numbers
- 1.07: Permutation Matrices
- 1.08: Projection Matrices
- 02: Systems of Linear Equations
- 2.01: Gaussian Elimination
- 2.02: When There is No Unique Solution
- 2.03: Reduced Row Echelon Form
- 2.04: Computing Inverses
- 2.05: LU Decomposition
- 03: Vector Spaces
- 3.01: Vector Spaces
- 3.02: Linear Independence
- 3.03: Span Basis and Dimension
- 3.04: Inner Product Spaces
- 3.05: Vector Spaces of a Matrix
- 3.06: Gram-Schmidt Process
- 3.07: Orthogonal Projections
- 3.08: QR Factorization
- 3.09: The Least-Squares Problem
- 3.10: 3. 10- Solution of the Least-Squares Problem
- 04: Determinants
- 4.01: Two-by-Two and Three-by-Three Determinants
- 4.02: Laplace Expansion and Leibniz Formula
- 4.03: Properties of the Determinant
- 4.04: Use of Determinants in Vector Calculus
- 05: Eigenvalues and Eigenvectors
- 5.01: The Eigenvalue Problem
- 5.02: Matrix Diagonalization
- 5.03: Symmetric and Hermitian Matrices
- 03: III. Differential Equations
- 06: Introduction to ODEs
- 6.01: The Simplest Type of Differential Equation
- 07: First-Order ODEs
- 7.01: The Euler Method
- 7.02: Separable Equations
- 7.03: Linear Equations
- 7.04: Applications
- 08: Second-Order ODEs Constant Coefficients
- 8.01: The Euler Method
- 8.02: The Principle of Superposition
- 8.03: The Wronskian
- 8.04: Homogeneous ODEs
- 8.05: Difference Equations
- 8.06: Inhomogenous ODEs
- 8.07: Resonance
- 8.08: Applications
- 8.09: Damped Resonance
- 09: Series Solutions
- 9.01: Ordinary Points
- 10: Systems of Linear Differential Equations
- 10.01: Distinct Real Eigenvalues
- 10.02: Solution by Diagonalization
- 10.03: Solution by the Matrix Exponential
- 10.04: Distinct Complex-Conjugate Eigenvalues
- 10.05: Repeated Eigenvalues with One Eigenvector
- 10.06: Normal Modes
- 11: Nonlinear Differential Equations
- 11.01: Fixed Points and Stability
- 11.02: Bifurcation Theory