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Differential equations and linear algebra are the two crucial courses in undergraduate mathematics. This new textbook develops those subjects separately and together. The complete book is a year's course, including Fourier and Laplace transforms, plus the Fast Fourier Transform and Singular Value Decomposition.
Undergraduate students in courses covering differential equations and linear algebra, either separately or together, will find this material essential to their understanding.
Preface
First Order Equations
Four Examples: Linear versus Nonlinear
The Calculus you Need
The Exponentials
Four Particular Solutions
Real and Complex Sinusoids
Models of Growth and Decay
The Logistic Equation
Separable Equations and Exact Equations
Second Order Equations
Second Derivatives in Science and Engineering
Key Facts About Complex Numbers
Constant Coefficients A, B, C
Forced Oscillations and Exponential Response
Electrical Networks and Mechanical Systems
Solutions to Second Order Equations
Laplace Transforms Y(s) and F(s)
Graphical and Numerical Methods
Nonlinear Equations
Sources, Sinks, Saddles, and Spirals
Linearization and Stability in 2D and 3D
The Basic Euler Methods
Higher Accuracy with Runge-Kutta
Linear Equations and Inverse Matrices
Two Pictures of Linear Equations
Solving Linear Equations by Elimination
Matrix Multiplication
Inverse Matrices
Symmetric Matrices and Orthogonal Matrices
Vector Spaces and Subspaces
The Column Space of a Matrix
The Nullspace of A: Solving Av=0
The Complete Solution to Av=b
Independence, Basis and Dimension
The Four Fundamental Subspaces
Graphs and Networks
Eigenvalues and Eigenvectors
Introduction to Eigenvalues
Diagonalizing a Matrix
Linear Systems y =Ay
The Exponential of a Matrix
Second Order Systems and Symmetric Matrices
Applied Mathematics and ATA
Least Squares and Projections
Positive Definite Matrices and the SVD
Boundary Conditions Replace Initial Conditions
Laplace's Equation
Networks and the Graph Laplacian
Fourier and Laplace Transforms
Fourier Series
The Fast Fourier Transform
The Heat Equation
The Wave Equation
The Laplace Transform
Convolution (Fourier and Laplace)
Matrix Factorizations
Properties of Determinants
Index
Linear Algebra in a Nutshell