# linear algebra and smarandache linear algebra

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The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and vector spaces over finite prime fields, which is not properly represented or analyzed in linear algebra books.TRANSCRIPT

w. b. vasantha kandasamy

LINEAR ALGEBRA AND

SMARANDACHE LINEAR ALGEBRA

AMERICAN RESEARCH PRESS

2003

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Linear Algebra and Smarandache Linear Algebra

W. B. Vasantha Kandasamy Department of Mathematics

Indian Institute of Technology, Madras Chennai 600036, India

American Research Press

2003

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This book can be ordered in a paper bound reprint from: Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA Tel.: 1-800-521-0600 (Customer Service)

http://wwwlib.umi.com/bod/

and online from: Publishing Online, Co. (Seattle, Washington State) at: http://PublishingOnline.com This book has been peer reviewed and recommended for publication by: Jean Dezert, Office National d=Etudes et de Recherches Aerospatiales (ONERA), 29, Avenue de la Division Leclerc, 92320 Chantillon, France. M. Khoshnevisan, School of Accounting and Finance, Griffith University, Gold Coast, Queensland 9726, Australia. Sabin Tabirca and Tatiana Tabirca, University College Cork, Cork, Ireland. Copyright 2003 by American Research Press and W. B. Vasantha Kandasamy Rehoboth, Box 141 NM 87322, USA Many books can be downloaded from our E-Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN: 1-931233-75-6 Standard Address Number: 297-5092 Printed in the United States of America

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CONTENTS

PREFACE 5

Chapter One LINEAR ALGEBRA : Theory and Applications

1.1 Definition of linear Algebra and its properties 7 1.2 Linear transformations and linear operators 12 1.3 Elementary canonical forms 20 1.4 Inner product spaces 29 1.5 Operators on inner product space 33 1.6 Vector spaces over finite fields Zp 37 1.7 Bilinear forms and its properties 44 1.8 Representation of finite groups 46 1.9 Semivector spaces and semilinear algebra 48

1.10 Some applications of linear algebra 60 Chapter Two SMARANDACHE LINEAR ALGEBRA AND ITS PROPERTIES

2.1 Definition of different types of Smarandache linear algebra with examples 65 2.2 Smarandache basis and S-linear transformation of S-vector spaces 71 2.3 Smarandache canonical forms 76 2.4 Smarandache vector spaces defined over finite S-rings Zn 81 2.5 Smarandache bilinear forms and its properties 86 2.6 Smarandache representation of finite S-semigroup 88 2.7 Smarandache special vector spaces 99 2.8 Algebra of S-linear operators 103 2.9 Miscellaneous properties in Smarandache linear algebra 110

2.10 Smarandache semivector spaces and Smarandache semilinear algebras 119

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Chapter Three SMARANDACHE LINEAR ALGEBRAS AND ITS APPLICATIONS

3.1 A smattering of neutrosophic logic using S-vector spaces of type II 141 3.2 Smarandache Markov Chains using S-vector spaces II 142 3.3 Smarandache Leontief economic models 143 3.4 Smarandache anti-linear algebra 146

Chapter Four SUGGESTED PROBLEMS 149

REFERENCES 165

INDEX 169

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PREFACE While I began researching for this book on linear algebra, I was a little startled. Though, it is an accepted phenomenon, that mathematicians are rarely the ones to react surprised, this serious search left me that way for a variety of reasons. First, several of the linear algebra books that my institute library stocked (and it is a really good library) were old and crumbly and dated as far back as 1913 with the most 'new' books only being the ones published in the 1960s. Next, of the few current and recent books that I could manage to find, all of them were intended only as introductory courses for the undergraduate students. Though the pages were crisp, the contents were diluted for the aid of the young learners, and because I needed a book for research-level purposes, my search at the library was futile. And given the fact, that for the past fifteen years, I have been teaching this subject to post-graduate students, this absence of recently published research level books only increased my astonishment. Finally, I surrendered to the world wide web, to the pulls of the internet, where although the results were mostly the same, there was a solace of sorts, for, I managed to get some monographs and research papers relevant to my interests. Most remarkable among my internet finds, was the book by Stephen Semmes, Some topics pertaining to the algebra of linear operators, made available by the Los Alamos National Laboratory's internet archives. Semmes' book written in November 2002 is original and markedly different from the others, it links the notion of representation of group and vector spaces and presents several new results in this direction. The present book, on Smarandache linear algebra, not only studies the Smarandache analogues of linear algebra and its applications, it also aims to bridge the need for new research topics pertaining to linear algebra, purely in the algebraic sense. We have introduced Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra and their fuzzy equivalents. Moreover, in this book, we have brought out the study of linear algebra and vector spaces over finite prime fields, which is not properly represented or analyzed in linear algebra books. This book is divided into four chapters. The first chapter is divided into ten sections which deal with, and introduce, all notions of linear algebra. In the second chapter, on Smarandache Linear Algebra, we provide the Smarandache analogues of the various concepts related to linear algebra. Chapter three suggests some application of Smarandache linear algebra. We indicate that Smarandache vector spaces of type II

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will be used in the study of neutrosophic logic and its applications to Markov chains and Leontief Economic models both of these research topics have intense industrial applications. The final chapter gives 131 significant problems of interest, and finding solutions to them will greatly increase the research carried out in Smarandache linear algebra and its applications. I want to thank my husband Dr.Kandasamy and two daughters Meena and Kama for their continued work towards the completion of these books. They spent a lot of their time, retiring at very late hours, just to ensure that the books were completed on time. The three of them did all the work relating to the typesetting and proofreading of the books, taking no outside help at all, either from my many students or friends. I also like to mention that this is the tenth and final book in this book series on Smarandache Algebraic Structures. I started writing these ten books, on April 14 last year (the prized occasion being the birth anniversary of Dr.Babasaheb Ambedkar), and after exactly a year's time, I have completed the ten titles. The whole thing would have remained an idle dream, but for the enthusiasm and inspiration from Dr. Minh Perez of the American Research Press. His emails, full of wisdom and an unbelievable sagacity, saved me from impending depression. When I once mailed him about the difficulties I am undergoing at my current workplace, and when I told him how my career was at crisis, owing to the lack of organizational recognition, it was Dr. Minh who wrote back to console me, adding: "keep yourself deep in research (because later the books and articles will count, not the titles of president of IIT or chair at IIT, etc.). The books and articles remain after our deaths." The consolation and prudent reasoning that I have received from him, have helped me find serenity despite the turbulent times in which I am living in. I am highly indebted to Dr. Minh for the encouragement and inspiration, and also for the comfort and consolation. Finally I dedicate this book to millions of followers of Periyar and Babasaheb Ambedkar. They rallied against the casteist hegemony prevalent at the institutes of research and higher education in our country, continuing in the tradition of the great stalwarts. They organized demonstrations and meetings, carried out extensive propaganda, and transformed the campaign against brahmincal domination into a people's protest. They spontaneously helped me, in every possible and imaginable way, in my crusade against the upper caste tyranny and domination in the Indian Institute of Technology, Madras -- a foremost bastion of the brahminical forces. The support they lent to me, while I was singlehandedly struggling, will be something that I shall cherish for the rest of my life. If I am a survivor today, it is because of their brave crusade for social justice.

W.B.Vasantha Kandasamy

14 April 2003

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Chapter One

LINEAR ALGEBRA Theory and Applications This chapter has ten sections, which tries to give a possible outlook on linear algebra. The notions given are basic concepts and results that are recalled without proof. The reader is expected to be well-acquainted with concepts in linear algebra to proceed on with this book. However chapter one helps for quick reference of basic concepts. In section one we give the definition and some of the properties of linear algebra. Linear transformations and linear operators are introduced in section two. Section three gives the basic concepts on canonical forms. Inner product spaces are dealt in section four and section five deals with forms and operator on inner product spaces. Section six is new for we do not have any book dealing separately with vector spaces built over finite fields Zp. Here it is completely introduced and analyzed. Section seven is devoted to the study and introduction of bilinear forms and its properties. Section eight is unconventional for

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