Linear Algebra and Differential Equations Using MATLAB

1. PRELIMINARIES Vectors and Matrices / MATLAB(R) / 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(R) / 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(R) 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(R) COMMANDS / ANSWERS TO SELECTED ODD-NUMBERED PROBLEMS / INDEX