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Introduction to Linear Bialgebra INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Florentin Smarandache K. Ilanthenral 2005 1 INTRODUCTION TO LINEAR BIALGEBRA W. B. Vasantha Kandasamy Department of Mathematics Indian Institute of Technology, Madras Chennai – 600036, India e-mail: [email protected] web: http://mat.iitm.ac.in/~wbv Florentin Smarandache Department of Mathematics University of New Mexico Gallup, NM 87301, USA e-mail: [email protected] K. Ilanthenral Editor, Maths Tiger, Quarterly Journal Flat No.11, Mayura Park, 16, Kazhikundram Main Road, Tharamani, Chennai – 600 113, India e-mail: [email protected] 2005 2 CONTENTS Preface 5 Chapter One INTRODUCTION TO LINEAR ALGEBRA AND S- LINEAR ALGEBRA 1.1 Basic properties of linear algebra 7 1.2 Introduction to s-linear algebra 15 1.3 Some aapplications of S-linear algebra 30 Chapter Two INTRODUCTORY COCEPTS OF BASIC BISTRUCTURES AND S-BISTRUCTURES 2.1 Basic concepts of bigroups and bivector spaces 37 2.2 Introduction of S-bigroups and S-bivector spaces 46 Chapter Three LINEAR BIALGEBRA, S-LINEAR BIALGEBRA AND THRIR PROPERTIES 3.1 Basic properties of linear bialgebra 51 3.2 Linear bitransformation and linear bioperators 62 3.3 Bivector spaces over finite fields 93 3.4 Representation of finite bigroup 95 3 3.5 Aaalications of bimatrix to bigraphs 102 3.6 Jordan biform 108 3.7 Application of bivector spaces to bicodes 113 3.8 Best biapproximation and its application 123 3.9 Markov bichains–biprocess 129 Chapter Four NEUTROSOPHIC LINEAR BIALGEBRA AND ITS APPLICATION 4.1 Some basic neutrosophic algebraic structures 131 4.2 Smarandache neutrosophic linear bialgebra 170 4.3 Smarandache representation of finite Smaradache bisemigroup 174 4.4 Smarandache Markov bichains using S- neutrosophic bivector spaces 189 4.5 Smandrache neutrosophic Leontif economic bimodels 191 Chapter Five SUGGESTED PROBLEMS 201 BIBLIOGRAPHY 223 INDEX 229 ABOUT THE AUTHORS 238 4 Preface The algebraic structure, linear algebra happens to be one of the subjects which yields itself to applications to several fields like coding or communication theory, Markov chains, representation of groups and graphs, Leontief economic models and so on. This book has for the first time, introduced a new algebraic structure called linear bialgebra, which is also a very powerful algebraic tool that can yield itself to applications. With the recent introduction of bimatrices (2005) we have ventured in this book to introduce new concepts like linear bialgebra and Smarandache neutrosophic linear bialgebra and also give the applications of these algebraic structures. It is important to mention here it is a matter of simple exercise to extend these to linear n-algebra for any n greater than 2; for n = 2 we get the linear bialgebra. This book has five chapters. In the first chapter we just introduce some basic notions of linear algebra and S- linear algebra and their applications. Chapter two introduces some new algebraic bistructures. In chapter three we introduce the notion of linear bialgebra and discuss several interesting properties about them. Also, application of linear bialgebra to bicodes is given. A remarkable part of our research in this book is the introduction of the notion of birepresentation of bigroups. The fourth chapter introduces several neutrosophic algebraic structures since they help in defining the new concept of neutrosophic linear bialgebra, neutrosophic bivector spaces, Smarandache neutrosophic linear bialgebra and Smarandache neutrosophic bivector spaces. Their 5 probable applications to real-world models are discussed. We have aimed to make this book engrossing and illustrative and supplemented it with nearly 150 examples. The final chapter gives 114 problems which will be a boon for the reader to understand and develop the subject. The main purpose of this book is to familiarize the reader with the applications of linear bialgebra to real-world problems. Finally, we express our heart-felt thanks to Dr.K.Kandasamy whose assistance and encouragement in every manner made this book possible. W.B.VASANTHA KANDASAMY FLORENTIN SMARANDACHE K. ILANTHENRAL 6 Chapter One INTRODUCTION TO LINEAR ALGEBRA AND S-LINEAR ALGEBRA In this chapter we just give a brief introduction to linear algebra and S-linear algebra and its applications. This chapter has three sections. In section one; we just recall the basic definition of linear algebra and some of the important theorems. In section two we give the definition of S-linear algebra and some of its basic properties. Section three gives a few applications of linear algebra and S-linear algebra. 1.1 Basic properties of linear algebra In this section we give the definition of linear algebra and just state the few important theorems like Cayley Hamilton theorem, Cyclic Decomposition Theorem, Generalized Cayley Hamilton Theorem and give some properties about linear algebra. 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 y α in V, called the product of c and α, in such a way that 1. 1y α = α for every α in V. 2. (c1 y c2)y α = c1 y (c2y α ). 3. c y (α + β) = cy α + cy β. 4. (c1 + c2)y α = c1y α + c2y α . for α, β ∈ V and c, c1 ∈ F. 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. V is a linear algebra if V has a multiplicative closed binary operation ‘.’ which is associative; i.e., if v1, v2 ∈ V, v1.v2 ∈ V. The same set of vectors may be part of a number of distinct vectors. 8 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. THEOREM (CAYLEY HAMILTON): Let T be a linear operator on a finite dimensional vector space V. If f is the characteristic polynomial for T, then f(T) = 0, in other words the minimal polynomial divides the characteristic polynomial for T. THEOREM: (CYCLIC DECOMPOSITION THEOREM): Let T be a linear operator on a finite dimensional vector space V and let W0 be a proper T-admissible subspace of V. There exist non-zero vectors α1, …, αr in V with respective T- annihilators p1, …, pr such that i. V = W0 ⊕ Z(α1; T) ⊕ … ⊕ Z (αr; T). ii. pt divides pt–1, t = 2, …, r. Further more the integer r and the annihilators p1, …, pr are uniquely determined by (i) and (ii) and the fact that αt is 0. THEOREM (GENERALIZED CAYLEY HAMILTON THEOREM): Let T be a linear operator on a finite dimensional vector space V. Let p and f be the minimal and characteristic polynomials for T, respectively i. p divides f. ii. p and f have the same prime factors except the multiplicities. α1 αt iii. If p = f1 … ft is the prime factorization of p, d1 d2 dt then f = f1 f2 … ft where di is the nullity of α fi(T) divided by the degree of fi . 9 The following results are direct and left for the reader to prove. Here we take vector spaces only over reals i.e., real numbers. We are not interested in the study of these properties in case of complex fields. Here we recall the concepts of linear functionals, adjoint, unitary operators and normal operators. DEFINITION 1.1.2: Let F be a field of reals and V be a vector space over F. An inner product on V is a function which assigns to each ordered pair of vectors α , β in V a scalar 〈α / β〉 in F in such a way that for all α, β, γ in V and for all scalars c. i. 〈α + β | γ〉 = 〈α | γ〉 + 〈β | γ〉. ii. 〈c α | β〉 = c〈α | β〉. iii. 〈β | α〉 = 〈α | β〉. iv. 〈α | α〉 > 0 if α ≠ 0. v. 〈α | cβ + γ〉 = c〈α | β〉 + 〈α | γ〉. Let Q n or F n be a n dimensional vector space over Q or F respectively for α, β ∈ Q n or F n where α = 〈α1, α2, …, αn〉 and β = 〈β1, β2, …, βn〉 〈α | β〉 = ∑α j β j . j Note: We denote the positive square root of 〈α | α〉 by ||α|| and ||α|| is called the norm of α with respect to the inner product 〈 〉. We just recall the notion of quadratic form. The quadratic form determined by the inner product is the function that assigns to each vector α the scalar ||α||2. Thus we call an inner product space is a real vector space together with a specified inner product in that space. A finite dimensional real inner product space is often called a Euclidean space. 10 The following result is straight forward and hence the proof is left for the reader.
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