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Introduction to Linear Algebra and Differential Equations

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Introduction to Linear Algebra and Differential Equations Cover

 

Synopses & Reviews

Publisher Comments:

Written for the undergraduate who has completed a year of calculus, this clear, skillfully organized text combines two important topics in modern mathematics in one comprehensive volume.

As Professor Dettman (Oakland University, Rochester, Michigan) points out, "Not only is linear algebra indispensable to the mathematics major, but . . . it is that part of algebra which is most useful in the application of mathematical analysis to other areas, e.g. linear programming, systems analysis, statistics, numerical analysis, combinatorics, and mathematical physics."

The book progresses from familiar ideas to more complex and difficult concepts, with applications introduced along the way, to clarify or illustrate theoretical material.

Among the topics covered are complex numbers, including two-dimensional vectors and functions of a complex variable; matrices and determinants; vector spaces; symmetric and hermitian matrices; first order nonlinear equations; linear differential equations; power-series methods; Laplace transforms; Bessel functions; systems of differential equations; and boundary value problems.

To reinforce and expand each chapter, numerous worked-out examples are included. A unique pedagogical feature is the starred section at the end of each chapter. Although these sections are not essential to the sequence of the book, they are related to the basic material and offer advanced topics to stimulate the more ambitious student. These topics include power series; existence and uniqueness theorems; Hilbert spaces; Jordan forms; Green's functions; Bernstein polynomials; and the Weierstrass approximation theorem.

This carefully structured textbook provides an ideal, step-by-step transition from first-year calculus to multivariable calculus and, at the same time, enables the instructor to offer special challenges to students ready for more advanced material.

Synopsis:

Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.

Synopsis:

Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.

Table of Contents

  Preface

1. Complex Numbers

  1.1 Introduction

  1.2 The Algebra of Complex Numbers

  1.3 The Geometry of Complex Numbers

  1.4 Two-dimensional Vectors

  1.5 Functions of a Complex Variable

  1.6 Exponential Function

  1.7 Power Series

2. Linear Algebraic Equations

  2.1 Introduction

  2.2 Matrices

  2.3 Elimination Method

  2.4 Determinants

  2.5 Inverse of a Matrix

  2.6 Existence and Uniqueness Theorems

3. Vector Spaces

  3.1 Introduction

  3.2 Three-dimensional Vectors

  3.3 Axioms of a Vector Space

  3.4 Dependence and Independence of Vectors

  3.5 Basis and Dimension

  3.6 Scalar Product

  3.7 Orthonormal Bases

  3.8 Infinite-dimensional Vector Spaces

4. Linear Transformations

  4.1 Introduction

  4.2 Definitions and Examples

  4.3 Matrix Representations

  4.4 Changes of Bases

  4.5 Characteristic Values and Characteristic Vectors

  4.6 Symmetric and Hermitian Matrices

  4.7 Jordan Forms

5. First Order Differential Equations

  5.1 Introduction

  5.2 An Example

  5.3 Basic Definitions

  5.4 First Order Linear Equations

  5.5 First Order Nonlinear Equations

  5.6 Applications of First Order Equations

  5.7 Numerical Methods

  5.8 Existence and Uniqueness

6. Linear Differential Equations

  6.1 Introduction

  6.2 General Theorems

  6.3 Variation of Parameters

  6.4 Equations with Constant Coefficients

  6.5 Method of Undetermined Coefficients

  6.6 Applications

  6.7 Green's Functions

7. Laplace Transforms

  7.1 Introduction

  7.2 Existence of the Transform

  7.3 Transforms of Certain Functions

  7.4 Inversion of the Transform

  7.5 Solution of Differential Equations

  7.6 Applications

  7.7 Uniqueness of the Transform

8. Power-Series Methods

  8.1 Introduction

  8.2 Solution near Ordinary Points

  8.3 Solution near Regular Singular Points

  8.4 Bessel Functions

  8.5 Boundary-value Problems

  8.6 Convergence Theorems

9. Systems of Differential Equations

  9.1 Introduction

  9.2 First Order Systems

  9.3 Linear First Order Systems

  9.4 Linear First Order Systems with Constant Coefficients

  9.5 Higher Order Linear Systems

  9.6 Existence and Uniqueness Theorem

  Answers and Hints for Selected Exercises; Index

Product Details

ISBN:
9780486651910
Author:
Dettman, John W.
Publisher:
Dover Publications
Author:
Dettman, John W.
Author:
Mathematics
Location:
New York :
Subject:
Differential Equations
Subject:
Algebra - General
Subject:
Algebra - Linear
Subject:
Algebras, linear
Subject:
General Mathematics
Subject:
Mathematics-Linear Algebra
Copyright:
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
no. 37
Publication Date:
19861231
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
416
Dimensions:
8.5 x 5.38 in 1.1 lb

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Related Subjects

Health and Self-Help » Health and Medicine » General
Health and Self-Help » Health and Medicine » General Medicine
Science and Mathematics » Mathematics » Algebra » General
Science and Mathematics » Mathematics » Algebra » Linear Algebra
Science and Mathematics » Mathematics » Differential Equations
Science and Mathematics » Mathematics » General

Introduction to Linear Algebra and Differential Equations New Trade Paper
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Product details 416 pages Dover Publications - English 9780486651910 Reviews:
"Synopsis" by ,
Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
"Synopsis" by , Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
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