Linear Algebra and Differential Equations Using MATLAB

by Golubitsky, Martin; Dellnitz, Michael

• ISBN13: 9780534354251
• ISBN10: 0534354254
• Edition: 1st
• Format: Hardcover
• Copyright: 1999-01-18
• Publisher: Cengage Learning

Summary

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

Preface	VIII
Preliminaries	1	(15)
Vectors and Matrices	1	(3)
MATLAB	4	(3)
Special Kinds of Matrices	7	(2)
The Geometry of Vector Operations	9	(7)
Solving Linear Equations	16	(45)
Systems of Linear Equations and Matrices	17	(7)
The Geometry of Low-Dimensional Solutions	24	(9)
Gaussian Elimination	33	(12)
Reduction to Echelon Form	45	(7)
Linear Equations with Special Coefficients	52	(6)
*Uniqueness of Reduced Echelon Form	58	(3)
Matrices and Linearity	61	(124)
Matrix Multiplication of Vectors	62	(5)
Matrix Mappings	67	(5)
Linearity	72	(7)
The Principle of Superposition	79	(4)
Composition and Multiplication of Matrices	83	(5)
Properties of Matrix Multiplication	88	(4)
Solving Linear Systems and Inverses	92	(9)
Determinants of 2 x 2 Matrices	101
Solving Ordinary Differential Equations	$$$
A Single Differential Equation	107	(12)
Graphing Solutions to Differential Equations	119	(7)
Phase Space Pictures and Equilibria	126	(7)
*Separation of Variables	133	(9)
Uncoupled Linear Systems of Two Equations	142	(7)
Coupled Linear Systems	149	(4)
The Initial Value Problem and Eigenvectors	153	(9)
Eigenvalues of 2 x 2 Matrices	162	(6)
Initial Value Problems Revisited	168	(7)
*Markov Chains	175	(10)
Vector Spaces	185	(33)
Vector Spaces and Subspaces	186	(6)
Construction of Subspaces	192	(6)
Spanning Sets and MATLAB	198	(3)
Linear Dependence and Linear Independence	201	(5)
Dimension and Bases	206	(5)
The Proof of the Main Theorem	211	(7)
Closed Form Solutions For Planar Odes	218	(45)
The Initial Value Problem	219	(6)
Closed Form Solutions by the Direct Method	225	(9)
Solutions Using Matrix Exponentials	234	(7)
Linear Normal Form Planar Systems	241	(3)
Similar Matrices	244	(7)
*Formulas for Matrix Exponentials	251	(5)
Second-Order Equations	256	(7)
Qualitative Theory of Planar Odes	263	(22)
Sinks, Saddles, and Sources	263	(7)
Phase Portraits of Sinks	270	(8)
Phase Portraits of Nonhyperbolic Systems	278	(7)
Determinants And Eigenvalues	285	(24)
Determinants	285	(12)
Eigenvalues	297	(8)
*Appendix: Existence of Determinants	305	(4)
Linear Maps And Changes Of Coordinates	309	(27)
Linear Mappings and Bases	309	(7)
Row Rank Equals Column Rank	316	(4)
Vectors and Matrices in Coordinates	320	(11)
Matrices of Linear Maps on a Vector Space	331	(5)
Orthogonality	336	(32)
Orthonormal Bases	336	(4)
Least Squares Approximations	340	(7)
Least Squares Fitting of Data	347	(9)
Symmetric Matrices	356	(4)
Orthogonal Matrices and QR Decompositions	360	(8)
Autonomous Planar Nonlinear Systems	368	(26)
Introduction	369	(6)
Equilibria and Linearization	375	(9)
Periodic Solutions	384	(6)
Stylized Phase Portraits	390	(4)
Bifurcation Theory	394	(46)
Two-Species Population Models	395	(7)
Examples of Bifurcations	402	(14)
The Continuous Flow Stirred Tank Reactor	416	(7)
The Remaining Global Bifurcations	423	(5)
*Saddle-Node Bifurcations Revisited	428	(6)
*Hopf Bifurcations Revisited	434	(6)
Matrix Normal Forms	440	(39)
Real Diagonalizable Matrices	440	(6)
Simple Complex Eigenvalues	446	(10)
Multiplicity and Generalized Eigenvectors	456	(6)
The Jordan Normal Form Theorem	462	(9)
*Appendix: Markov Matrix Theory	471	(4)
*Appendix: Proof of Jordan Normal Form	475	(4)
Higher Dimensional Systems	479	(45)
Linear Systems in Jordan Normal Form	480	(9)
Qualitative Theory near Equilibria	489	(6)
MATLAB ode45 in One Dimension	495	(8)
Higher Dimensional Systems Using ode45	503	(6)
Quasiperiodic Motions and Tori	509	(7)
Chaos and the Lorenz Equation	516	(8)
Linear Differential Equations	524	(40)
Solving Systems in Original Coordinates	525	(12)
Higher Order Equations	537	(7)
Linear Differential Operators	544	(6)
Undetermined Coefficients	550	(7)
Periodic Forcing and Resonance	557	(7)
Laplace Transforms	564	(75)
The Method of Laplace Transforms	565	(6)
Laplace Transforms and Their Computation	571	(8)
Partial Fractions	579	(5)
Discontinuous Forcing	584	(6)
RLC Circuits	590
Additional Techniques for Solving Odes	$$$
Nonconstant Coefficient Linear Equations	597	(5)
Variation of Parameters for Systems	602	(6)
The Wronskian	608	(3)
Higher Order Equations	611	(8)
Simplification by Substitution	619	(4)
Exact Differential Equations	623	(10)
Hamiltonian Systems	633	(6)
Numerical Solutions of Odes	639	(30)
A Description of Numerical Methods	640	(12)
Error Bounds for Euler's Method	652	(6)
Local and Global Error Bounds	658	(9)
Appendix: Variable Step Methods	667	(2)
MATLAB Commands	669	(6)
Answers to Selected Odd-Numbered Problems	675	(22)
Index	697