These world-renowned authors integrate linear algebra and ordinary differential equations in this unique book, interweaving instructions on how to use MATLAB® with examples and theory. They use computers in two ways: in linear algebra, computers reduce the drudgery of calculations to help students focus on concepts and methods; in differential equations, computers display phase portraits graphically for students to focus on the qualitative information embodied in solutions, rather than just to learn to develop formulas for solutions.

### Table of Contents

1. PRELIMINARIES

Vectors and Matrices / MATLAB® / Special Kinds of Matrices / The Geometry of Vector Operations

2. SOLVING LINEAR EQUATIONS

Systems of Linear Equations and Matrices / The Geometry of Low-Dimensional Solutions / Gaussian Elimination / Reduction to Echelon Form / Linear Equations with Special Coefficients / Uniqueness of Reduced Echelon Form

3. MATRICES AND LINEARITY

Matrix Multiplication of Vectors / Matrix Mappings / Linearity / The Principle of Superposition / Composite and Multiplication of Matrices / Properties of Matrix Multiplication / Solving Linear Systems and Inverses / Determinants of 2 x 2 Matrices

4. SOLVING ORDINARY DIFFERENTIAL EQUATIONS

A Single Differential Equation / Graphing Solutions to Differential Equations / Phase Space Pictures and Equilibria / Separation of Variables / Uncoupled Linear Systems of Two Equations / Coupled Linear Systems / The Initial Value Problem and Eigenvectors / Eigenvalues of 2 x 2 Matrices / Initial Value Problems Revisited / Markov Chains

5. VECTOR SPACES

Vector Spaces and Subspaces / Construction of Subspaces / Spanning Sets and MATLAB® / Linear Dependence and Linear Independence / Dimension and Bases / The Proof of the Main Theorem

6. CLOSED FORM SOLUTIONS FOR PLANAR ODES

The Initial Value Problem / Closed Form Solutions by the Direct Method / Solutions Using Matrix Exponentials / Linear Normal Form Planar Systems / Similar Matrices / Formulas for Matrix Exponentials / Second Order Equations

7. QUALITATIVE THEORY OF PLANAR ODES

Sinks, Saddles, and Sources / Phase Portraits of Sinks / Phase Portraits of Nonhyperbolic Systems

8. DETERMINANTS AND EIGENVALUES

Determinants / Eigenvalues / Appendix: Existence of Determinants

9. LINEAR MAPS AND CHANGES OF COORDINATES

Linear Mappings and Bases / Row Rank Equals Column Rank / Vectors and Matrices in Coordinates / Matrices of Linear Maps on a Vector Space

10. ORTHOGONALITY

Orthonormal Bases / Least Squares Approximations / Least Squares Fitting of Data / Symmetric Matrices / Orthogonal Matrices of QR Decompositions

11. AUTONOMOUS PLANAR NONLINEAR SYSTEMS

Introduction / Equilibria and Linearization / Periodic Solutions / Stylized Phase Portraits

12. BIFURCATION THEORY

Two Species Population Models / Examples of Bifurcations / The Continuous Flow Stirred Tank Reactor / The Remaining Global Bifurcations / Saddle-Node Bifurcations Revisited / Hopf Bifurcations Revisited

13. MATRIX NORMAL FORMS

Real Diagonalizable Matrices / Simple Complex Eigenvalues / Multiplicity and Generalized Eigenvectors / The Jordan Normal Form Theorem / Appendix: Markov Matrix Theory / Appendix: Proof of Jordan Normal Form

14. HIGHER DIMENSIONAL SYSTEMS

Linear Systems in Jordan Normal Form / Qualitative Theory Near Equilibria / MATLAB® ODE45 in One Dimension / Higher Dimensional Systems Using ODE45 / Quasiperiodic Motions and Tori / Chaos and the Lorenz Equation

15. LINEAR DIFFERENTIAL EQUATIONS

Solving Systems in Original Coordinates / Higher Order Equations / Linear Differential Operators / Undetermined Coefficients / Periodic Forcing and Resonance

16. LAPLACE TRANSFORMS

The Method of Laplace Transforms / Laplace Transforms and Their Computation / Partial Fractions / Discontinuous Forcing / RLC Circuits

17. ADDITIONAL TECHNIQUES FOR SOLVING ODES

Nonconstant Coefficient Linear Equations / Variation of Parameters for Systems / The Wronskian / Higher Order Equations / Simplification by Substitution / Exact Differential Equations / Hamiltonian Systems

18. NUMERICAL SOLUTIONS OF ODES

A Description of Numerical Methods / Error Bounds for Euler's Method / Local and Global Error Bounds / APPENDIX: VARIABLE STEP METHODS / MATLAB® COMMANDS / ANSWERS TO SELECTED ODD-NUMBERED PROBLEMS / INDEX