Linear Algebra and Differential Equations Using MATLAB

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.

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

Martin Golubitsky Michael Dellnitz Golubitsky 

Informatik EDV Textbooks Mathematics / Algebra / Linear Informatik/EDV 

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