Algebra and Analysis for Engineers and Scientists

__"This is an intermediate level text, with exercises, whose avowed purpose is to provide the science and engineering graduate student with an appropriate modern mathematical (analysis and algebra) background in a succinct, but nontrivial, manner.... [T]he book is quite thorough and can serve as a text, for self-study, or as a reference."__ **—Mathematical Reviews** Written for graduate and advanced undergraduate students in engineering and science, this classic book focuses primarily on set theory, algebra, and analysis. Useful as a course textbook, for self-study, or as a reference, the work is intended to: \* provide readers with appropriate mathematical background for graduate study in engineering or science; \* allow students in engineering or science to become familiar with a great deal of pertinent mathematics in a rapid and efficient manner without sacrificing rigor; \* give readers a unified overview of applicable mathematics, enabling them to choose additional, advanced topical courses in mathematics more intelligently. Whereas these objectives for writing this book were certainly pertinent over twenty years ago when the work was first published, they are even more compelling now. Today’s graduate students in engineering or science are expected to be more knowledgeable and sophisticated in mathematics than students in the past. Moreover, today’s graduate students in engineering or science are expected to be familiar with a great deal of ancillary material (primarily in the computer science area), acquired in courses that did not even exist a couple of decades ago. The book is divided into three parts: __set theory__ (Chapter 1), __algebra__ (Chapters 2–4), and __analysis__ (Chapters 5–7). The first two chapters deal with the fundamental concepts of sets, functions, relations and equivalence relations, and algebraic structures. Chapters 3 and 4 cover vector spaces and linear transformations, and finite-dimensional vector spaces and matrices. The last three chapters investigate metric spaces, normed and inner product spaces, and linear operators. Because of its flexible structure, **Algebra and Analysis for Engineers and Scientists may be used either in a one- or two-semester course by deleting appropriate sections, taking into account the students’ backgrounds and interests.** A generous number of exercises have been integrated into the text, and a section of references and notes is provided at the end of each chapter. Applications of algebra and analysis having a broad appeal are also featured, including topics dealing with ordinary differential equations, integral equations, applications of the contraction mapping principle, minimization of functionals, an example from optimal control, and estimation of random variables. Supplementary material for students and instructors is available at http://Michel.Herget.net. This book evolved from a one-year sequence of courses offered by the authors at Iowa State University. The audience for this book typically included theoreti­ cally oriented first- or second-year graduate students in various engineering or science disciplines. Subsequently, while serving as Chair of the Department of Electrical Engineering, and later, as Dean of the College of Engineering at the University of Notre Dame, the first author continued using this book in courses aimed primarily at graduate students in control systems. Since administrative demands precluded the possibility of regularly scheduled classes, the Socratic method was used in guiding students in self study. This method of course deliv­ ery turned out to be very effective and satisfying to student and teacher alike. Feedback from colleagues and students suggests that this book has been used in a similar manner elsewhere. The original objectives in writing this book were to provide the reader with ap­ propriate mathematical background for graduate study in engineering or science; to provide the reader with appropriate prerequisites for more advanced subjects in mathematics; to allow the student in engineering or science to become famil­ iar with a great deal of pertinent mathematics in a rapid and efficient manner without sacrificing rigor; to give the reader a unified overview of applicable mathematics, thus enabling him or her to choose additional courses in math­ ematics more intelligently; and to make it possible for the student to understand Cover Page......Page 1 REVIEWS OF Algebra and Analysis for Engineers and Scientists......Page 2 Title: Algebra and Analysis for Engineers and Scientists......Page 5 ISBN 0817647066......Page 6 CONTENTS......Page 7 PREFACE......Page 11 Suggested Course Outlines......Page 13 1 FUNDAMENTAL CONCEPTS......Page 17 2 ALGEBRAIC STRUCTURES......Page 49 3 VECTOR SPACES AND LINEAR TRANSFORMATIONS......Page 91 4 FINITE-DIMENSIONAL VECTOR SPACES AND MATRICES......Page 140 5 METRIC SPACES......Page 279 6 NORMED SPACES AND INNER PRODUCT SPACES......Page 359 7 LINEAR OPERATORS......Page 422 A......Page 491 B,C......Page 492 F......Page 493 I......Page 494 K,L......Page 495 O......Page 496 P,Q,R......Page 497 S......Page 498 U......Page 499 V,W,X,Y,Z......Page 500 Written for graduate and advanced undergraduate students in engineering and science, this classic book focuses primarily on set theory, algebra, and analysis. Useful as a course textbook, for self-study, or as a reference, the work is intended to familiarize engineering and science students with a great deal of pertinent and applicable mathematics in a rapid and efficient manner without sacrificing rigor. The book is divided into three parts: set theory, algebra, and analysis. It offers a generous number of exercises integrated into the text and features applications of algebra and analysis that have a broad appeal.