Undergraduate students in courses covering differential equations and linear algebra, either separately or together, will find this material essential to their understanding.

**Contents:**Preface; 1: First Order Equations; 1.1: Four
Examples: Linear versus Nonlinear; 1.2: The Calculus you Need;
1.3: The Exponentials; 1.4: Four Particular Solutions; 1.5: Real
and Complex Sinusoids; 1.6: Models of Growth and Decay; 1.7: The
Logistic Equation; 1.8: Separable Equations and Exact Equations;
2: Second Order Equations; 2.1: Second Derivatives in Science and
Engineering; 2.2: Key Facts About Complex Numbers; 2.3: Constant
Coefficients *A, B, C*; 2.4: Forced Oscillations and
Exponential Response; 2.5: Electrical Networks and Mechanical
Systems; 2.6: Solutions to Second Order Equations; 2.7: Laplace
Transforms *Y*(s) and *F*(s); 3: Graphical and
Numerical Methods; 3.1: Nonlinear Equations; 3.2: Sources, Sinks,
Saddles, and Spirals; 3.3: Linearization and Stability in 2D and
3D; 3.4: The Basic Euler Methods; 3.5: Higher Accuracy with
Runge-Kutta; 4: Linear Equations and Inverse Matrices; 4.1: Two
Pictures of Linear Equations; 4.2: Solving Linear Equations by
Elimination; 4.3: Matrix Multiplication; 4.4: Inverse Matrices;
4.5: Symmetric Matrices and Orthogonal Matrices; 5: Vector Spaces
and Subspaces; 5.1: The Column Space of a Matrix; 5.2: The
Nullspace of *A*: Solving *Av=0*; 5.3: The Complete
Solution to *Av=b*; 5.4: Independence, Basis and Dimension;
5.5: The Four Fundamental Subspaces; 5.6: Graphs and Networks; 6:
Eigenvalues and Eigenvectors; 6.1: Introduction to Eigenvalues;
6.2: Diagonalizing a Matrix; 6.3: Linear Systems *y*
=*Ay*; 6.4: The Exponential of a Matrix; 6.5: Second Order
Systems and Symmetric Matrices; 7: Applied Mathematics and ATA;
7.1: Least Squares and Projections; 7.2: Positive Definite
Matrices and the SVD; 7.3: Boundary Conditions Replace Initial
Conditions; 7.4: Laplace's Equation; 7.5: Networks and the Graph
Laplacian; 8: Fourier and Laplace Transforms; 8.1: Fourier
Series; 8.2: The Fast Fourier Transform; 8.3: The Heat Equation;
8.4: The Wave Equation; 8.5: The Laplace Transform; 8.6:
Convolution (Fourier and Laplace); Matrix Factorizations;
Properties of Determinants; Index; Linear Algebra in a Nutshell.