Preface

1Preliminaries

1.1Vectors and Matrices

1.2MATLAB

1.3Special Kinds of Matrices

1.4The Geometry of Vector Operations

2Solving Linear Equations

2.1

2.2The Geometry of Low-Dimensional Solutions

2.3

2.4

2.5Linear Equations with Special Coefficients

2.6Uniqueness of Reduced Echelon Form

3Matrices and Linearity

3.1

3.2Matrix Mappings

3.3

3.4The Principle of Superposition

3.5

3.6

3.7

3.8Determinants of 2 × 2 Matrices

4Solving Ordinary Differential Equations

4.1A Single Differential Equation

4.2Graphing Solutions to Differential Equations

4.3Phase Space Pictures and Equilibria

4.4Separation of Variables

4.5Uncoupled Linear Systems of Two Equations

4.6Coupled Linear Systems

4.7The Initial Value Problem and Eigenvectors

4.8Eigenvalues of 2 × 2 Matrices

4.9Initial Value Problems Revisited

4.10

5Vector Spaces

5.1Vector Spaces and Subspaces

5.2Construction of Subspaces

5.3

5.4

5.5Dimension and Bases

5.6

6Closed Form Solutions for Planar ODEs

6.1The Initial Value Problem

6.2Closed Form Solutions by the Direct Method

6.3Solutions Using Matrix Exponentials

6.4Linear Normal Form Planar Systems

6.5Similar Matrices

6.6Formulas for Matrix Exponentials

6.7Second Order Equations

7Qualitative Theory of Planar ODEs

7.1Sinks, Saddles, and Sources

7.2Phase Portraits of Sinks

7.3Phase Portraits of Nonhyperbolic Systems

8Determinants and Eigenvalues

8.1Determinants

8.2Eigenvalues

8.3Existence of Determinants

9Linear Maps and Changes of Coordinates

9.1Linear Mappings and Bases

9.2

9.3

9.4

10Orthogonality

10.1

10.2

10.3Least Squares Fitting of Data

10.4Symmetric Matrices

10.5

11Matrix Normal Forms

11.1

11.2

11.3

11.4

11.5Markov Matrix Theory

11.6Proof of Jordan Normal Form

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