Orthogonal Sets and Polar Methods in Linear

A unique, applied approach to problem solving in linear algebra Departing from the standard methods of analysis, this unique book presents methodologies and algorithms based on the concept of orthogonality and demonstrates their application to both standard and novel problems in linear algebra. Covering basic theory of linear systems, linear inequalities, and linear programming, it focuses on elegant, computationally simple solutions to real-world physical, economic, and engineering problems. The authors clearly explain the reasons behind the analysis of different structures and concepts and use numerous illustrative examples to correlate the mathematical models to the reality they represent. Readers are given precise guidelines for: * Checking the equivalence of two systems * Solving a system in certain selected variables * Modifying systems of equations * Solving linear systems of inequalities * Using the new exterior point method * Modifying a linear programming problem With few prerequisites, but with plenty of figures and tables, end-of-chapter exercises as well as Java and Mathematica programs available from the authors' Web site, this is an invaluable text/reference for mathematicians, engineers, applied scientists, and graduate students in mathematics.

ENRIQUE CASTILLO is a professor in the Department of Applied Mathematics and Computational Science at the University of Cantabria, Spain. During 25 years of research and teaching, he has published hundreds of papers as well as 18 books. ANGEL COBO is an associate professor in the Department of Applied Mathematics and Computational Science in the University of Cantabria.<br> <br> FRANCISCO JUBETE is a civil engineer and research assistant in the Department of Applied Mathematics and Computational Science at the University of Cantabria.<br> <br> VA PRUNEDA is a research assistant in the Department of Applied Mathematics and Computational Sciences at the University of Cantabria.

Part I Linear Spaces and Systems of Equations

Basic Concepts

5

(18)

Introduction

5

(1)

Linear space

6

(5)

The Euclidean Space En

11

(3)

Orthogonal Sets and Decompositions

14

(2)

Matrices

16

(4)

Systems of Linear Equations

20

(1)

Exercises

21

(2)

Orthogonal Sets

23

(18)

Introduction and Motivation

23

(1)

Orthogonal Decompositions

24

(7)

The Orthogonalization Module

31

(5)

Mathematica Program

36

(3)

Exercises

39

(2)

Matrix Calculations Using Orthogonal Sets

41

(32)

Introduction

41

(1)

Inverting a Matrix

42

(1)

The Rank of a Matrix

43

(1)

Calculating the Determinant of a Matrix

44

(2)

Algorithm for Matrix Calculations

46

(4)

Complexity

50

(1)

Inverses and Determinants of Row-Modified Matrices

51

(4)

Inverses of Symbolic Matrices

55

(1)

Extensions to Partitioned Matrices

56

(4)

Inverses of Modified Matrices

60

(7)

Mathematica Programs

67

(2)

Exercises

69

(4)

More Applications of Orthogonal Sets

73

(12)

Intersection of Two Linear Subspaces

73

(4)

Reciprocals Images in Linear Transformations

77

(2)

Other Applications

79

(2)

Mathematica Programs

81

(2)

Exercises

83

(2)

Orthogonal Sets and Systems of Linear Equations

85

(62)

Introduction

85

(1)

Compatibility of a System of Linear Equations

86

(4)

Solving a System of Linear Equations

90

(5)

Complexity

95

(1)

Checking Systems Equivalence

96

(3)

Solving a System in Some Selected Variables

99

(2)

Modifying Systems of Equations

101

(23)

Applications

124

(11)

Mathematica Programs

135

(2)

Exercises

137

(10)

Appendix: Proof of Lemma 5.2

140

(7)

Part II Cones and Systems of Inequalities

Polyhedral Convex Cones

147

(44)

Introduction

147

(1)

Convex Sets

147

(1)

Types of Linear Combinations

148

(3)

Polyhedral Convex Cones

151

(6)

The r-Process

157

(25)

The Complete r-Algorithm

182

(2)

Mathematica Program

184

(5)

Exercises

189

(2)

Polytopes and Polyhedra

191

(24)

Introduction

191

(1)

Polytopes

191

(16)

Polyhedra

207

(5)

Exercises

212

(3)

Cones and Systems of Inequalities

215

(40)

Introduction

215

(1)

A Discussion of Linear Systems

216

(10)

Solving Linear Systems

226

(21)

Applications to Linear Programming

247

(1)

Exercises

248

(7)

Part III Linear Programming

An Introduction to Linear Programming

255

(20)

Introduction

255

(1)

Problem Statement and Basic Definitions

255

(6)

Linear Programming Problem in Standard Form

261

(3)

Basic Solutions

264

(1)

Duality

264

(7)

Exercises

271

(4)

The Exterior Point Method

275

(70)

Introduction

275

(1)

The Exterior Point Method

275

(24)

Making the EPM More Efficient

299

(19)

Complexity

318

(2)

Recovering the Final Tableau from the Solution

320

(1)

Modifying a Linear Programming Problem

321

(18)

Exercises

339

(6)

Part IV Applications

Applications

345

(50)

Introduction

345

(1)

Matrix Analysis of Engineering Structures

346

(19)

The Transportation Problem

365

(4)

Production--Scheduling Problems

369

(7)

The Input--Output Tables

376

(2)

The Diet Problem

378

(2)

Network Flow Problems

380

(7)

Exercises

387

(8)

Part V Appendices

Appendix A: A Java Application

395

(1)

How to Use the Program

396

(11)

Appendix B: List of Notation

407

(8)

References

415

(4)

Index

419

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.

FREE SHIPPING

Amazon no longer offers textbook rentals. We do!

Orthogonal Sets and Polar Methods in Linear Algebra Applications to Matrix Calculations, Systems of Equations, Inequalities, and Linear Programming

Summary

Author Biography

Table of Contents

Supplemental Materials