Linear Algebra and Differential Equations

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.

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

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Linear Algebra and Differential Equations Using MATLAB

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Table of Contents

Supplemental Materials