Introduction To Linear Bialgebra

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INTRODUCTIONTO LINEAR BIALGEBRAW. B. Vasantha KandasamyDepartment of MathematicsIndian Institute of Technology, MadrasChennai – 600036, Indiae-mail: vasantha@iitm.ac.inweb: http://mat.iitm.ac.in/ wbvFlorentin SmarandacheDepartment of MathematicsUniversity of New MexicoGallup, NM 87301, USAe-mail: smarand@gallup.unm.eduK. IlanthenralEditor, Maths Tiger, Quarterly JournalFlat No.11, Mayura Park,16, Kazhikundram Main Road, Tharamani,Chennai – 600 113, Indiae-mail: ilanthenral@gmail.comHEXISPhoenix, Arizona2005

This book can be ordered in a paper bound reprint from:Books on DemandProQuest Information & Learning(University of Microfilm International)300 N. Zeeb RoadP.O. Box 1346, Ann ArborMI 48106-1346, USATel.: 1-800-521-0600 (Customer Service)http://wwwlib.umi.com/bod/and online from:Publishing Online, Co. (Seattle, Washington State)at: http://PublishingOnline.comThis book has been peer reviewed and recommended for publication by:Dr. Jean Dezert, Office National d'Etudes et de Recherches Aerospatiales(ONERA), 29, Avenue de la Division Leclerc, 92320 Chantillon, France.Dr. Iustin Priescu, Academia Technica Militaria, Bucharest, Romania,Prof. Dr. B. S. Kirangi, Department of Mathematics and Computer Science,University of Mysore, Karnataka, India.Copyright 2005 by W. B. Vasantha Kandasamy, Florentin Smarandache andK. IlanthenralLayout by Kama Kandasamy.Cover design by Meena Kandasamy.Many books can be downloaded from:http://www.gallup.unm.edu/ smarandache/eBooks-otherformats.htmISBN: 1931233-97-7Standard Address Number: 297-5092Printed in the United States of America

CONTENTS5PrefaceChapter OneINTRODUCTION TO LINEAR ALGEBRA AND SLINEAR ALGEBRA1.11.21.3Basic properties of linear algebraIntroduction to s-linear algebraSome aapplications of S-linear algebra71530Chapter TwoINTRODUCTORY COCEPTS OF BASICBISTRUCTURES AND S-BISTRUCTURES2.12.2Basic concepts of bigroups and bivectorspacesIntroduction of S-bigroups and S-bivectorspaces3746Chapter ThreeLINEAR BIALGEBRA, S-LINEAR BIALGEBRA ANDTHRIR PROPERTIES3.13.23.33.4Basic properties of linear bialgebra51Linear bitransformation and linear bioperators 62Bivector spaces over finite fields93Representation of finite bigroup953

3.53.63.73.83.9Applications of bimatrix to bigraphsJordan biformApplication of bivector spaces to bicodesBest biapproximation and its applicationMarkov bichains–biprocess102108113123129Chapter FourNEUTROSOPHIC LINEAR BIALGEBRA AND ITSAPPLICATION4.14.24.34.44.5Some basic neutrosophic algebraicstructuresSmarandache neutrosophic linear bialgebraSmarandache representation of finiteSmarandache bisemigroupSmarandache Markov bichains using Sneutrosophic bivector spacesSmarandache neutrosophic Leontifeconomic bimodels131170174189191Chapter FiveSUGGESTED PROBLEMS201BIBLIOGRAPHY223INDEX229ABOUT THE AUTHORS2384

PrefaceThe algebraic structure, linear algebra happens to be one ofthe subjects which yields itself to applications to severalfields like coding or communication theory, Markov chains,representation of groups and graphs, Leontief economicmodels 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 yielditself to applications.With the recent introduction of bimatrices (2005)we have ventured in this book to introduce new conceptslike linear bialgebra and Smarandache neutrosophic linearbialgebra and also give the applications of these algebraicstructures.It is important to mention here it is a matter ofsimple exercise to extend these to linear n-algebra for any ngreater than 2; for n 2 we get the linear bialgebra.This book has five chapters. In the first chapter wejust introduce some basic notions of linear algebra and Slinear algebra and their applications. Chapter two introducessome new algebraic bistructures. In chapter three weintroduce the notion of linear bialgebra and discuss severalinteresting properties about them. Also, application of linearbialgebra to bicodes is given. A remarkable part of ourresearch in this book is the introduction of the notion ofbirepresentation of bigroups.The fourth chapter introduces several neutrosophicalgebraic structures since they help in defining the newconcept of neutrosophic linear bialgebra, neutrosophicbivector spaces, Smarandache neutrosophic linear bialgebraand Smarandache neutrosophic bivector spaces. Their5

probable applications to real-world models are discussed.We have aimed to make this book engrossing andillustrative and supplemented it with nearly 150 examples.The final chapter gives 114 problems which will be a boonfor the reader to understand and develop the subject.The main purpose of this book is to familiarize thereader with the applications of linear bialgebra to real-worldproblems.Finally, we express our heart-felt thanks toDr.K.Kandasamy whose assistance and encouragement inevery manner made this book possible.W.B.VASANTHA KANDASAMYFLORENTIN SMARANDACHEK. ILANTHENRAL6

Chapter OneINTRODUCTION TOLINEAR ALGEBRAAND S-LINEAR ALGEBRAIn this chapter we just give a brief introduction to linearalgebra and S-linear algebra and its applications. Thischapter has three sections. In section one; we just recall thebasic definition of linear algebra and some of the importanttheorems. In section two we give the definition of S-linearalgebra and some of its basic properties. Section three givesa few applications of linear algebra and S-linear algebra.1.1 Basic properties of linear algebraIn this section we give the definition of linear algebra andjust state the few important theorems like Cayley Hamiltontheorem, Cyclic Decomposition Theorem, GeneralizedCayley Hamilton Theorem and give some properties aboutlinear algebra.DEFINITION 1.1.1: A vector space or a linear spaceconsists of the following:i.ii.a field F of scalars.a set V of objects called vectors.7

iii.a rule (or operation) called vector addition; whichassociates with each pair of vectors α, β V; α β in V, called the sum of α and β in such a way thata. 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 zerovector, such thatα 0 αfor all α in V.d. for each vector α in V there is a unique vector –α in V such thatα (–α) 0.e. a rule (or operation), called scalarmultiplication, which associates with eachscalar c in F and a vector α in V a vector c y αin V, called the product of c and α, in such away that1. 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 vectorspace 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 binaryoperation ‘.’ which is associative; i.e., if v1, v2 V, v1.v2 V. The same set of vectors may be part of a number ofdistinct vectors.8

We simply by default of notation just say V a vectorspace over the field F and call elements of V as vectors onlyas matter of convenience for the vectors in V may not bearmuch resemblance to any pre-assigned concept of vector,which the reader has.THEOREM (CAYLEY HAMILTON): Let T be a linearoperator on a finite dimensional vector space V. If f is thecharacteristic polynomial for T, then f(T) 0, in otherwords the minimal polynomial divides the characteristicpolynomial for T.THEOREM: (CYCLIC DECOMPOSITION THEOREM): Let Tbe a linear operator on a finite dimensional vector space Vand let W0 be a proper T-admissible subspace of V. Thereexist non-zero vectors α1, , αr in V with respective Tannihilators p1, , pr such thati.ii.V W0 Z(α1; T) Z (αr; T).pt divides pt–1, t 2, , r.Further more the integer r and the annihilators p1, , prare uniquely determined by(i) and (ii) and the fact that αt is 0.THEOREM(GENERALIZEDCAYLEYHAMILTONTHEOREM): Let T be a linear operator on a finitedimensional vector space V. Let p and f be the minimal andcharacteristic polynomials for T, respectivelyi.ii.p divides f.p and f have the same prime factors except themultiplicities.iii.If p f1 1 f t t is the prime factorization of p,ααdddthen f f1 1 f 2 2 f t t where di is the nullity offi(T) α divided by the degree of fi .9

The following results are direct and left for the reader toprove.Here we take vector spaces only over reals i.e., realnumbers. We are not interested in the study of theseproperties in case of complex fields. Here we recall theconcepts of linear functionals, adjoint, unitary operators andnormal operators.DEFINITION 1.1.2: Let F be a field of reals and V be avector space over F. An inner product on V is a functionwhich assigns to each ordered pair of vectors α , β in V ascalar 〈α / β〉 in F in such a way that for all α, β, γ in V andfor all scalars c.i.ii.iii.iv.v.〈α β γ〉 〈α γ〉 〈β γ〉.〈c α β〉 c〈α β〉.〈β α〉 〈α β〉.〈α α〉 0 if α 0.〈α cβ γ〉 c〈α β〉 〈α γ〉.Let Q n or F n be a n dimensional vector space over Q or Frespectively for α, β Q n or F n whereα 〈α1, α2, , αn〉 andβ 〈β1, β2, , βn〉〈α β〉 α j β j .jNote: We denote the positive square root of 〈α α〉 by α and α is called the norm of α with respect to the innerproduct 〈 〉.We just recall the notion of quadratic form.The quadratic form determined by the inner product isthe function that assigns to each vector α the scalar α 2.Thus we call an inner product space is a real vectorspace together with a specified inner product in that space.A finite dimensional real inner product space is often calleda Euclidean space.10

The following result is straight forward and hence theproof is left for the reader.Result 1.1.1: If V is an inner product space, then for anyvectors α, β in V and any scalar c.i.ii.iii.iv. cα c α . α 0 for α 0. 〈α β〉 α β . α β α β .Let α and β be vectors in an inner product space V. Then αis orthogonal to β if 〈α β〉 0 since this implies β isorthogonal to α, we often simply say that α and β areorthogonal. If S is a set of vectors in V, S is called anorthogonal set provided all pair of distinct vectors in S areorthogonal. An orthogonal set S is an orthonormal set if itsatisfies the additional property α 1 for every α in S.Result 1.1.2: An orthogonal set of non-zero vectors islinearly independent.Result 1.1.3: If α and β is a linear combination of anorthogonal sequence of non-zero vectors α1, , αm then βis the particular linear combinationsm〈β α t 〉αt .2t 1 α t β Result 1.1.4: Let V be an inner product space and let β1, ,βn be any independent vectors in V. Then one may constructorthogonal vectors α1, ,αn in V such that for each t 1, 2, , n the set {α1, , αt} is a basis for the subspace spannedby β1, , βt.This result is known as the Gram-Schmidtorthgonalization process.11

Result 1.1.5: Every finite dimensional inner product spacehas an orthogonal basis.One of the nice applications is the concept of a bestapproximation. A best approximation to β by vector in W isa vector α in W such that β – α β – γ for every vector γ in W.The following is an important concept relating to thebest approximation.THEOREM 1.1.1: Let W be a subspace of an inner productspace V and let β be a vector in V.i.The vector α in W is a best approximation to βby vectors in W if and only if β – α is orthogonalto every vector in W.ii.If a best approximation to β by vectors in Wexists it is unique.iii.If W is finite dimensional and {α1, , αt } is anyorthogonal basis for W, then the vectorα t( β α t )α t α t 2is the unique best approximation to β by vectors inW.Let V be an inner product space and S any set of vectors inV. The orthogonal complement of S is that set S of allvectors in V which are orthogonal to every vector in S.Whenever the vector α exists it is called the orthogonalprojection of β on W. If every vector in V has orthogonalprojection on W, the mapping that assigns to each vector in12

V its orthogonal projection on W is called the orthogonalprojection of V on W.Result 1.1.6: Let V be an inner product space, W is a finitedimensional subspace and E the orthogonal projection of Von W.Then the mappingβ β – Eβis the orthogonal projection of V on W .Result 1.1.7: Let W be a finite dimensional subspace of aninner product space V and let E be the orthogonal projectionof V on W. Then E is an idempotent linear transformationof V onto W, W is the null space of E and V W W .Further 1 – E is the orthogonal projection of V on W . It isan idempotent linear transformation of V onto W with nullspace W.Result 1.1.8: Let {α1, , αt} be an orthogonal set of nonzero vectors in an inner product space V.If β is any vector in V, then (β, α t ) 2 α 2 β 2ttand equality holds if and only ifβ (β α t )α .2 tt αtNow we prove the existence of adjoint of a linear operator Ton V, this being a linear operator T such that (Tα β) (α T β) for all α and β in V.We just recall some of the essential results in thisdirection.13

Result 1.1.9: Let V be a finite dimensional inner productspace and f a linear functional on V. Then there exists aunique vector β in V such that f(α) (α β) for all α in V.Result 1.1.10: For any linear operator T on a finitedimensional inner product space V there exists a uniquelinear operator T on V such that(Tα β) (α T β)for all α, β in V.Result 1.1.11: Let V be a finite dimensional inner productspace and let B {α1, , αn} be an (ordered) orthonormalbasis for V. Let T be a linear operator on V and let A be thematrix of T in the ordered basis B. ThenAij (Tαj αi).Now we define adjoint of T on V.DEFINITION 1.1.3: Let T be a linear operator on an innerproduct space V. Then we say that T has an adjoint on V ifthere exists a linear operator T on V such that(Tα β) (α T β)for all α, β in V.It is important to note that the adjoint of T depends not onlyon T but on the inner product as well.The nature of T is depicted by the following result.THEOREM 1.1.2: Let V be a finite dimensional innerproduct real vector space. If T and U are linear operatorson V and c is a scalari.(T U) T U .14

ii.iii.iv.(cT) cT .(TU) U T .(T ) T.A linear operator T such that T T is called self adjoint orHermitian.Results relating the orthogonal basis is left for the reader toexplore.Let V be a finite dimensional inner product space and T alinear operator on V. We say that T is normal if it commuteswith its adjoint i.e. TT T T.1.2 Introduction to S-linear algebraIn this section we first recall the definition of SmarandacheR-module and Smarandache k-vectorial space. Then wegive different types of Smarandache linear algebra andSmarandache vector space.Further we define Smarandache vector spaces over thefinite rings which are analogous to vector spaces definedover the prime field Zp. Throughout this section Zn willdenote the ring of integers modulo n, n a composite numberZn[x] will denote the polynomial ring in the variable x withcoefficients from Zn.DEFINITION [27, 40]: The Smarandache-R-Module (S-Rmodule) is defined to be an R-module (A, , ) such that aproper subset of A is a S-algebra (with respect with thesame induced operations and another ‘ ’ operation internalon A), where R is a commutative unitary Smarandache ring(S-ring) and S its proper subset that is a field. By a propersubset we understand a set included in A, different from theempty set, from the unit element if any and from A.DEFINITION [27, 40]: The Smarandache k-vectorial space(S-k-vectorial space) is defined to be a k-vectorial space,15

(A, , y) such that a proper subset of A is a k-algebra (withrespect with the same induced operations and another ‘ ’operation internal on A) where k is a commutative field. Bya proper subset we understand a set included in A differentfrom the empty set from the unit element if any and from A.This S-k-vectorial space will be known as type I, S-kvectorial space.Now we proceed on to define the notion of Smarandache kvectorial subspace.DEFINITION 1.2.1: Let A be a k-vectorial space. A propersubset X of A is said to be a Smarandache k-vectorialsubspace (S-k-vectorial subspace) of A if X itself is aSmarandache k-vectorial space.THEOREM 1.2.1: Let A be a k-vectorial space. If A has a Sk-vectorial subspace then A is a S-k-vectorial space.Proof: Direct by the very definitions.Now we proceed on to define the concept of Smarandachebasis for a k-vectorial space.DEFINITION 1.2.2: Let V be a finite dimensional vectorspace over a field k. Let B {ν1, ν 2 , , νn } be a basis ofV. We say B is a Smarandache basis (S-basis) of V if B hasa proper subset say A, A B and A φ and A B such thatA generates a subspace which is a linear algebra over kthat is if W is the subspace generated by A then W must be ak-algebra with the same operations of V.THEOREM 1.2.2: Let V be a vector space over the field k. IfB is a S-basis then B is a basis of V.Proof: Straightforward by the very definitions.The concept of S-basis leads to the notion of Smarandachestrong basis which is not present in the usual vector spaces.16

DEFINITION 1.2.3: Let V be a finite dimensional vectorspace over a field k. Let B {x1, , xn} be a basis of V. Ifevery proper subset of B generates a linear algebra over kthen we call B a Smarandache strong basis (S-strong basis)for V.Now having defined the notion of S-basis and S-strongbasis we proceed on to define the concept of Smarandachedimension.DEFINITION 1.2.4: Let L be any vector space over the fieldk. We say L is a Smarandache finite dimensional vectorspace (S-finite dimensional vector space) of k if every Sbasis has only finite number of elements in it. It isinteresting to note that if L is a finite dimensional vectorspace then L is a S-finite dimensional space provided L hasa S-basis.It can also happen that L need not be a finite dimensionalspace still L can be a S-finite dimensional space.THEOREM 1.2.3: Let V be a vector space over the field k. IfA {x1, , xn} is a S-strong basis of V then A is a S-basis ofV.Proof: Direct by definitions, hence left for the reader as anexercise.THEOREM 1.2.4: Let V be a vector space over the field k. IfA {x1, , xn } is a S-basis of V. A need not in general be aS-strong basis of V.Proof: By an example. Let V Q [x] be the set of allpolynomials of degree less than or equal to 10. V is a vectorspace over Q.Clearly A {1, x, x2, , x10 } is a basis of V. In fact Ais a S-basis of V for take B {1, x2, x4, x6, x8, x10}. ClearlyB generates a linear algebra. But all subsets of A do not17

form a S-basis of V, so A is not a S-strong basis of V butonly a S-basis of V.We will define Smarandache eigen values andSmarandache eigen vectors of a vector space.DEFINITION 1.2.5: Let V be a vector space over the field Fand let T be a linear operator from V to V. T is said to be aSmarandache linear operator (S-linear operator) on V if Vhas a S-basis, which is mapped by T onto anotherSmarandache basis of V.DEFINITION 1.2.6: Let T be a S-linear operator defined onthe space V. A characteristic value c in F of T is said to be aSmarandache characteristic value (S-characteristic value)of T if the characteristic vector of T associated with cgenerate a subspace, which is a linear algebra that is thecharacteristic space, associated with c is a linear algebra.So the eigen vector associated with the S-characteristicvalues will be called as Smarandache eigen vectors (S-eigenvectors) or Smarandache characteristic vectors (Scharacteristic vectors).Thus this is the first time such Smarandache notions aregiven about S-basis, S-characteristic values and Scharacteristic vectors. For more about these please refer [43,46].Now we proceed on to define type II Smarandache kvector spaces.DEFINITION 1.2.7: Let R be a S-ring. V be a module over R.We say V is a Smarandache vector space of type II (S-vectorspace of type II) if V is a vector space over a proper subsetk of R where k is a field.We have no means to interrelate type I and type IISmarandache vector spaces.However in case of S-vector spaces of type II we definea stronger version.18

DEFINITION 1.2.8: Let R be a S-ring, M a R-module. If M isa vector space over every proper subset k of R which is afield; then we call M a Smarandache strong vector space oftype II (S-strong vector space of type II).THEOREM 1.2.5: Every S-strong vector space of type II is aS-vector space of type II.Proof: Direct by the very definition.Example 1.2.1: Let Z12 [x] be a module over the S-ring Z12.Z12 [x] is a S-strong vector space of type II.Example 1.2.2: Let M2 2 {(aij) aij Z6} be the set of all2 2 matrices with entries from Z6. M2 2 is a module overZ6 and M2 2 is a S-strong vector space of type II.Example 1.2.3: Let M3 5 {(aij) aij Z6} be a moduleover Z6. Clearly M3 5 is a S-strong vector space of type IIover Z6.Now we proceed on to define Smarandache linearalgebra of type II.DEFINITION 1.2.9: Let R be any S-ring. M a R-module. M issaid to be a Smarandache linear algebra of type II (S-linearalgebra of type II) if M is a linear algebra over a propersubset k in R where k is a field.THEOREM 1.2.6: All S-linear algebra of type II is a Svector space of type II and not conversely.Proof: Let M be an R-module over a S-ring R. Suppose Mis a S-linear algebra II over k R (k a field contained in R)then by the very definition M is a S-vector space II.To prove converse we have show that if M is a S-vectorspace II over k R (R a S-ring and k a field in R) then M ingeneral need not be a S-linear algebra II over k contained inR. Now by example 1.2.3 we see the collection M3 5 is a S-19

vector space II over the field k {0, 2, 4} contained in Z6. Butclearly M3 5 is not a S-linear algebra II over {0, 2, 4} Z6.We proceed on to define Smarandache subspace II andSmarandache subalgebra II.DEFINITION 1.2.10: Let M be an R-module over a S-ring R.If a proper subset P of M is such that P is a S-vector spaceof type II over a proper subset k of R where k is a field thenwe call P a Smarandache subspace II (S-subspace II) of Mrelative to P.It is important to note that even if M is a R-module over aS-ring R, and M has a S-subspace II still M need not be a Svector space of type II.On similar lines we will define the notion ofSmarandache subalgebra II.DEFINITION 1.2.11: Let M be an R-module over a S-ring R.If M has a proper subset P such that P is a Smarandachelinear algebra II (S-linear algebra II) over a proper subsetk in R where k is a field then we say P is a S-linearsubalgebra II over R.Here also it is pertinent to mention that if M is a R-modulehaving a subset P that is a S-linear subalgebra II then Mneed not in general be a S-linear algebra II. It has becomeimportant to mention that in all algebraic structure, S if ithas a proper substructure P that is Smarandache then S itselfis a Smarandache algebraic structure. But we see in case ofR-Modules M over the S-ring R if M has a S-subspace or Ssubalgebra over a proper subset k of R where k is a fieldstill M in general need not be a S-vector space or a S-linearalgebra over k; k R.Now we will illustrate this by the following examples.Example 1.2.4: Let M R[x] R[x] be a direct product ofpolynomial rings, over the ring R R. Clearly M R[x] R[x] is a S-vector space over the field k R {0}.20

It is still more interesting to see that M is a S-vector spaceover k {0} Q, Q the field of rationals. Further M is a Sstrong vector space as M is a vector space over every propersubset of R R which is a field.Still it is important to note that M R [x] R [x] is a Sstrong linear algebra. We see Q[x] Q[x] P M is a Ssubspace over k1 Q {0} and {0} Q but P is not a Ssubspace over k2 R {0} or {0} R.Now we will proceed on to define Smarandache vectorspaces and Smarandache linear algebras of type III.DEFINITION 1.2.12: Let M be a any non empty set which isa group under ‘ ’. R any S-ring. M in general need not be amodule over R but a part of it is related to a section of R.We say M is a Smarandache vector space of type III (Svector space III) if M has a non-empty proper subset Pwhich is a group under ' ', and R has a proper subset ksuch that k is a field and P is a vector space over k.Thus this S-vector space III links or relates and gets a nicealgebraic structure in a very revolutionary way.We illustrate this by an example.Example 1.2.5: Consider M Q [x] Z [x]. Clearly M isan additively closed group. Take R Q Q; R is a S-ring.Now P Q [x] {0} is a vector space over k Q {0}.Thus we see M is a Smarandache vector space of type III.So this definition will help in practical problems whereanalysis is to take place in such set up.Now we can define Smarandache linear algebra of typeIII in an analogous way.DEFINITION 1.2.13: Suppose M is a S-vector space III overthe S-ring R. We call M a Smarandache linear algebra oftype III (S-linear algebra of type III) if P M which is avector space over k R (k a field) is a linear algebra.Thus we have the following naturally extended theorem.21

THEOREM 1.2.7: Let M be a S-linear algebra III for P Mover R related to the subfield k R. Then clearly P is a Svector space III.Proof: Straightforward by the very definitions.To prove that all S-vector space III in general is not a Slinear algebra III we illustrate by an example.Example 1.2.6: Let M P1 P2 where P1 M3 3 {(aij) aij Q} and P2 M2 2 {(aij) aij Z} and R be the fieldof reals. Now take the proper subset P P1, P1 is a S-vectorspace III over Q R. Clearly P1 is not a S-linear algebra III over Q.Now we proceed on to define S-subspace III and Slinear algebra III.DEFINITION 1.2.14: Let M be an additive Aeolian group, Rany S-ring. P M be a S-vector space III over a field k R. We say a proper subset T P to be a Smarandachevector subspace III (S-vector subspace III) or S-subspace IIIif T itself is a vector space over k.If a S-vector space III has no proper S-subspaces IIIrelative to a field k R then we call M a Smarandachesimple vector space III (S-simple vector space III).On similar lines one defines Smarandache sublinear algebraIII and S-simple linear algebra III.Yet a new notion called Smarandache super vectorspaces are introduced for the first time.DEFINITION 1.2.15: Let R be S-ring. V a module over R. Wesay V is a Smarandache super vector space (S-super vectorspace) if V is a S-k-vector space over a proper set k, k Rsuch that k is a field.THEOREM 1.2.8: All S-super spaces are S-k-vector spacesover the field k, k contained in R.22

Proof: Straightforward.Almost all results derived in case of S-vector spaces type IIcan also be derived for S-super vector spaces.Further for given V, a R-module of a S-ring R we canhave several S-super vector spaces.Now we just give the definition of Smarandache superlinear algebra.DEFINITION 1.2.16: Let R be a S-ring. V a R module overR. Suppose V is a S-super vector space over the field k, k R. we say V is a S-super linear algebra if for all a, b V wehave a y b V where 'y' is a closed associative binaryoperation on V.Almost all results in case of S-linear algebras can be easilyextended and studied in case of S-super linear algebras.DEFINITION 1.2.17: Let V be an additive abelian group, Znbe a S-ring (n a composite number). V is said to be aSmarandache vector space over Zn (S-vector space over Zn)if for some proper subset T of Zn where T is a fieldsatisfying the following conditions:i. vt , tv V for all v V and t T.ii. t (v1 v2) tv1 tv2 for all v1 v2 V and t T.iii. (t1 t2 ) v t1 v t2 v for all v V and t1 , t2 T.iv. t1 (t2 u) (t1 t2) u for all t1, t2 T and u V.v. if e is the identity element of the field T then ve ev v for all v V.In addition to all these if we have an multiplicativeoperation on V such that uy v1 V for all uy v1 V then wecall V a Smarandache linear algebra (S-linear algebra)defined over finite S-rings.23

It is a matter of routine to check that if V is a S-linearalgebra then obviously V is a S-vector space. We howeverwill illustrate by an example that all S-vector spaces ingeneral need not be S-linear algebras.Example 1.2.7: Let Z6 {0, 1, 2, 3, 4, 5} be a S-ring (ringof integers modulo 6). Let V M2 3 {(aij) aij Z6}.Clearly V is a S-vector space over T {0, 3}. But V isnot a S-linear algebra. Clearly V is a S-vector space over T1 {0, 2, 4}. The unit being 4 as 42 4 (mod 6).Example 1.2.8: Let Z12 {0, 1, 2, , 10, 11} be the S-ringof characteristic two. Consider the polynomial ring Z12[x] .Clearly Z12[x] is a S-vector space over the field k {0, 4, 8}where 4 is the identity element of k and k is isomorphicwith the prime field Z3.Example 1.2.9: Let Z18 {0, 1, 2, , 17} be the S-ring.M2 2 {(aij) aij Z18} M2 2 is a finite S-vector space overthe field k {0, 9}. What is the basis for such space?Here we see M2 2 has basis 1 0 0 1 0 0 0 0 0 0 , 0 0 , 0 1 and 1 0 . Clearly M2x2 is not a vector space over Z18 as Z18 is only aring.Now we proceed on to characterize those finite S-vectorspaces, which has only one field over which the space isdefined. We call such S-vector spaces as Smarandacheunital vector spaces. The S-vector space M2 2 defined overZ18 is a S-unital vector space. When the S-vector space hasmore than one S-vector space defined over more than onefield we call the S-vector space as Smarandache multivector space (S-multi vector space).For consider the vector space Z6[x]. Z6[x] is thepolynomial ring in the indeterminate x with coefficients24

from Z6. Clearly Z6[x] is a S-vector space over . k {0, 3};k is a field isomorphic with Z2 and Z6[x] is also a S-vectorspace over k1 {0, 2, 4} a field isomorphic to Z3 . ThusZ6[x] is called S-multi vector space.Now we have to

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-BISTRUCTU

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