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Set Algebra - Cover:Layout 1 6/17/2008 11:08 AM Page 1 SET LINEAR ALGEBRA AND SET FUZZY LINEAR ALGEBRA W. B. Vasantha Kandasamy e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv www.vasantha.net Florentin Smarandache e-mail: [email protected] K Ilanthenral e-mail: [email protected] INFOLEARNQUEST Ann Arbor 2008 1 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/ Peer reviewers: Prof. Dr. Adel Helmy Phillips. Faculty of Engineering, Ain Shams University 1 El-Sarayat st., Abbasia, 11517, Cairo, Egypt. Professor Mircea Eugen Selariu, Polytech University of Timisoara, Romania. Professor Paul P. Wang, Ph D Department of Electrical & Computer Engineering Pratt School of Engineering, Duke University Durham, NC 27708, USA Copyright 2008 by InfoLearnQuest and authors Cover Design and Layout by Kama Kandasamy Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks-otherformats.htm ISBN-10: 1-59973-029-4 ISBN-13: 978-1-59973-029-5 EAN: 9781599730295 Standard Address Number: 297-5092 Printed in the United States of America 2 CONTENTS Preface 5 Chapter One BASIC CONCEPTS 7 1.1 Definition of Linear Algebra and its Properties 7 1.2 Basic Properties of Linear Bialgebra 15 1.3 Fuzzy Vector Spaces 26 Chapter Two SET VECTOR SPACES 29 2.1 Set Vector Spaces and their Properties 30 2.2 Set Linear transformation of Set Vector Spaces 45 2.3 Set Linear Algebra and its Properties 50 3 2.4 Semigroup Vector spaces and their generalizations 55 2.5 Group Linear Algebras 86 Chapter Three SET FUZZY LINEAR ALGEBRAS AND THEIR PROPERTIES 119 Chapter Four SET BIVECTOR SPACES AND THEIR GENERALIZATION 133 Chapter Five SET n-VECTOR SPACES AND THEIR GENERALIZATIONS 175 Chapter Six SET FUZZY LINEAR BIALGEBRA AND ITS GENERALIZATION 237 Chapter Seven SUGGESTED PROBLEMS 283 REFERENCES 332 INDEX 337 ABOUT THE AUTHORS 344 4 PREFACE In this book, the authors define the new notion of set vector spaces which is the most generalized form of vector spaces. Set vector spaces make use of the least number of algebraic operations, therefore, even a non-mathematician is comfortable working with it. It is with the passage of time, that we can think of set linear algebras as a paradigm shift from linear algebras. Here, the authors have also given the fuzzy parallels of these new classes of set linear algebras. This book abounds with examples to enable the reader to understand these new concepts easily. Laborious theorems and proofs are avoided to make this book approachable for non- mathematicians. The concepts introduced in this book can be easily put to use by coding theorists, cryptologists, computer scientists, and socio-scientists. Another special feature of this book is the final chapter containing 304 problems. The authors have suggested so many problems to make the students and researchers obtain a better grasp of the subject. This book is divided into seven chapters. The first chapter briefly recalls some of the basic concepts in order to make this book self-contained. Chapter two introduces the notion of set vector spaces which is the most generalized concept of vector spaces. Set vector spaces lends itself to define new classes of vector spaces like semigroup vector spaces and group vector 5 spaces. These are also generalization of vector spaces. The fuzzy analogue of these concepts are given in Chapter three. In Chapter four, set vector spaces are generalized to biset bivector spaces and not set vector spaces. This is done taking into account the advanced information technology age in which we live. As mathematicians, we have to realize that our computer-dominated world needs special types of sets and algebraic structures. Set n-vector spaces and their generalizations are carried out in Chapter five. Fuzzy n-set vector spaces are introduced in the sixth chapter. The seventh chapter suggests more than three hundred problems. When a researcher sets forth to solve them, she/he will certainly gain a deeper understanding of these new notions. Our thanks are due to Dr. K. Kandasamy for proof-reading this book. We also acknowledge our gratitude to Kama and Meena for their help with the corrections and layout. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K.ILANTHENRAL 6 Chapter One BASIC CONCEPTS This chapter has three sections. In section one a brief introduction to linear algebra is given. Section two gives the basic notions in bilinear algebra and the final section gives the definition of fuzzy vector spaces. For more about these concepts, please refer [48, 60]. 1.1 Definition of Linear Algebra and its Properties In this section we just recall the definition of linear algebra and enumerate some of its basic properties. We expect the reader to be well versed with the concepts of groups, rings, fields and matrices. For these concepts will not be recalled in this section. Throughout this section, V will denote the vector space over F where F is any field of characteristic zero. DEFINITION 1.1.1: A vector space or a linear space consists of the following: i. a field F of scalars. ii. a set V of objects called vectors. 7 iii. a rule (or operation) called vector addition; which associates with each pair of vectors α, β ∈ V; α + β in V, called the sum of α and β in such a way that a. addition is commutative α + β = β + α. b. addition is associative α + (β + γ) = (α + β) + γ. c. there is a unique vector 0 in V, called the zero vector, such that α + 0 = α for all α in V. d. for each vector α in V there is a unique vector – α in V such that α + (–α) = 0. e. a rule (or operation), called scalar multiplication, which associates with each scalar c in F and a vector α in V, a vector c (cid:121) α in V, called the product of c and α, in such a way that 1. 1(cid:121) α = α for every α in V. 2. (c (cid:121) c )(cid:121) α = c (cid:121) (c(cid:121) α ). 1 2 1 2 3. c (cid:121) (α + β) = c(cid:121) α + c(cid:121) β. 4. (c + c )(cid:121) α = c(cid:121) α + c(cid:121) α . 1 2 1 2 for α, β ∈ V and c, c ∈ F. 1 It is important to note as the definition states that a vector space is a composite object consisting of a field, a set of ‘vectors’ and two operations with certain special properties. The same set of vectors may be part of a number of distinct vectors. We simply by default of notation just say V a vector space over the field F and call elements of V as vectors only as matter of convenience for the vectors in V may not bear much resemblance to any pre-assigned concept of vector, which the reader has. 8 Example 1.1.1: Let R be the field of reals. R[x] the ring of polynomials. R[x] is a vector space over R. R[x] is also a vector space over the field of rationals Q. Example 1.1.2: Let Q[x] be the ring of polynomials over the rational field Q. Q[x] is a vector space over Q, but Q[x] is clearly not a vector space over the field of reals R or the complex field C. Example 1.1.3: Consider the set V = R × R × R. V is a vector space over R. V is also a vector space over Q but V is not a vector space over C. Example 1.1.4: Let M = {(a ) ⏐ a ∈ Q} be the collection of m × n ij ij all m × n matrices with entries from Q. M is a vector space m × n over Q but M is not a vector space over R or C. m × n Example 1.1.5: Let ⎧⎛a a a ⎞ ⎫ 11 12 13 ⎪⎜ ⎟ ⎪ P3 × 3 = ⎨⎜a21 a22 a23⎟ aij∈Q,1≤i≤3, 1≤ j≤3 ⎬. ⎪⎜ ⎟ ⎪ ⎩⎝a a a ⎠ ⎭ 31 32 33 P is a vector space over Q. 3 × 3 Example 1.1.6: Let Q be the field of rationals and G any group. The group ring, QG is a vector space over Q. Remark: All group rings KG of any group G over any field K are vector spaces over the field K. We just recall the notions of linear combination of vectors in a vector space V over a field F. A vector β in V is said to be a linear combination of vectors ν …,ν in V provided there exists 1, n scalars c ,…, c in F such that 1 n 9