Linear Algebra and Differential Equations

by Peterson, Gary L.; Sochacki, James S.

• ISBN13: 9780201662122
• ISBN10: 0201662124
• Edition: 1st
• Format: Paperback
• Copyright: 2001-07-24
• Publisher: Pearson

Summary

This book has been written for a one-semester combined linear algebra and differential equations course, yet it contains enough material for a two-term sequence in linear algebra and differential equations. By introducing matrices, determinants, and vector spaces early in the course, the authors are able to fully develop the connections between linear algebra and differential equations. The book is flexible enough to be easily adapted to fit most syllabi, including courses that cover differential equations first. Technology is fully integrated where appropriate, and the text offers fresh and relevant applications to motivate student interest. Matrices and Determinants; Vector Spaces; First Order Ordinary Differential Equations; Linear Differential Equations; Linear Transformations and Eigenvalues and Eigenvectors; Systems of Differential Equations; The Laplace Transform; Power Series Solutions to Linear Differential Equations; Inner Product Spaces For all readers interested in linear algebra and differential equations.

Table of Contents

Matrices and Determinants	1	(64)
Systems of Linear Equations	2	(15)
Matrices and Matrix Operations	17	(11)
Inverses of Matrices	28	(9)
Special Matrices and Additional Properties of Matrices	37	(6)
Determinants	43	(8)
Further Properties of Determinants	51	(7)
Proofs of Theorems on Determinants	58	(7)
Vector Spaces	65	(46)
Vector Spaces	66	(8)
Subspaces and Spanning Sets	74	(9)
Linear Independence and Bases	83	(12)
Dimension; Nullspace, Row Space, and Column Space	95	(11)
Wronskians	106	(5)
First Order Ordinary Differential Equations	111	(68)
Introduction to Differential Equations	112	(8)
Separable Differential Equations	120	(4)
Exact Differential Equations	124	(6)
Linear Differential Equations	130	(6)
More Techniques for Solving First Order Differential Equations	136	(8)
Modeling with Differential Equations	144	(9)
Reduction of Order	153	(4)
The Theory of First Order Differential Equations	157	(11)
Numerical Solutions of Ordinary Differential Equations	168	(11)
Linear Differential Equations	179	(52)
The Theory of Higher Order Linear Differential Equations	179	(10)
Homogeneous Constant Coefficient Linear Differential Equations	189	(14)
The Method of Undetermined Coefficients	203	(8)
The Method of Variation of Parameters	211	(6)
Some Applications of Higher Order Differential Equations	217	(14)
Linear Transformations and Eigenvalues and Eigenvectors	231	(62)
Linear Transformations	231	(14)
The Algebra of Linear Transformations; Differential Operators and Differential Equations	245	(8)
Matrices for Linear Transformations	253	(16)
Eigenvalues and Eigenvectors of Matrices	269	(9)
Similar Matrices, Diagonalization, and Jordan Canonical Form	278	(9)
Eigenvectors and Eigenvalues of Linear Transformations	287	(6)
Systems of Differential Equations	293	(52)
The Theory of Systems of Linear Differential Equations	295	(7)
Homogeneous Systems with Constant Coefficients: The Diagonalizable Case	302	(10)
Homogeneous Systems with Constant Coefficients: The Nondiagonalizable Case	312	(3)
Nonhomogeneous Linear Systems	315	(4)
Converting Differential Equations to First Order Systems	319	(3)
Applications Involving Systems of Linear Differential Equations	322	(12)
2 x 2 Systems of Nonlinear Differential Equations	334	(11)
The Laplace Transform	345	(30)
Definition and Properties of the Laplace Transform	345	(7)
Solving Constant Coefficient Linear Initial Value Problems with Laplace Transforms	352	(4)
Step Functions, Impulse Functions, and the Delta Function	356	(10)
Convolution Integrals	366	(4)
Systems of Linear Differential Equations	370	(5)
Power Series Solutions to Linear Differential Equations	375	(38)
Introduction to Power Series Solutions	376	(8)
Series Solutions for Second Order Linear Differential Equations	384	(9)
Euler Type Equations	393	(4)
Series Solutions Near a Regular Singular Point	397	(16)
Inner Product Spaces	413	(28)
Inner Product Spaces	413	(8)
Orthonormal Bases	421	(9)
Schur's Theorem and Symmetric Matrices	430	(11)
Answers to Odd-Numbered Exercises	441	(18)
Index of Maple Commands	459	(2)
Index	461