Linear Algebra and Differential Equations Using MATLAB / Edition 1

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Overview

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.

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Editorial Reviews

From the Publisher
"The topic areas of linear algebra and differential equations lend themselves naturally to be developed in tandem, and I do not know of a text that does a better job of integrating the two areas so that each reinforce the understanding of the other."

"[This text] provides a very interesting way of blending linear algebra and sophomore level differential equations course with the programming system MATLAB. . . . It . . . has the potential to be adopted by many institutions who are leaning towards the use of technology in the classroom. For one, I would use the book fore our linear algebra and differential equations course. . . ."

"The text is unique in offering a true amalgam of linear algebra, differential equations, and MATLAB. There is a definite need for such a text. The integration of the three 'legs' is well done and the ideas are introduced at a good pace."

Booknews
Provides an integrated approach to linear algebra and ordinary differential equations based on computers. Includes instructions on how to use MATLAB, theories and examples, and preloaded matrices and differential equations with notes. Topics covered include: solving linear equations; matrices and linearity; solving ordinary differential equations; vector spaces; qualitative theory of planar odes; orthogonality; bifurcation theory; matrix normal forms; higher dimensional systems; and numerical solutions of odes. Also included are MATLAB commands and answers to selected problems. Annotation c. by Book News, Inc., Portland, Or.
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Product Details

  • ISBN-13: 9780534354251
  • Publisher: Cengage Learning
  • Publication date: 1/18/1999
  • Series: Mathematics Series
  • Edition description: New Edition
  • Edition number: 1
  • Pages: 720
  • Product dimensions: 7.40 (w) x 9.30 (h) x 1.20 (d)

Meet the Author

Professor of Mathematics at University of Houston, Ph.D. in Mathematics from Massachusetts Institute of Technology

Professor of Mathematics at University of Paderborn, Germany

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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

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