2. Basic Concepts
In this section basic concepts on vector spaces and a few of its properties and some NQ algebraic structures and their properties needed for this paper are given.
Through out this paper
R denotes the field of reals,
C denotes the field of complex numbers and
denotes the finite field of characteristic
p,
p a prime.
denotes the Neutrosophic Quadruple; with
in
R or
C or
, where
and
F has the usual neutrosophic logic meaning of Truth, Indeterminate and False respectively and
a denotes the known part [
26].
For basic properties of vector spaces and linear algebras please refer [
22].
Definition 1 ([
22])
. A vector space or a linear space V consists of the following;- 1.
A field of R or C or of scalars.
- 2.
A set V of objects called vectors.
- 3.
A rule (or operation) called vector addition; which associates with each pair of vectors in is in V, called sum of the vectors x and y in such a way that ;
- (a)
(addition is commutative).
- (b)
(addition is associative).
- (c)
There is a unique vector 0 in V such that for all .
- (d)
For each vector there is a unique vector such that
- (e)
A rule or operation called scalar multiplication that associates with each scalar or C or and for a vector , called product denoted by `.’ of c and x in such a way that for and and ;
- i.
for every .
- ii.
- iii.
- iv.
for all and in R or C or .
We can just say is a vector space over a field R or C or if is an additive abelian group and V is compatible with the product by the scalars. If on V is defined a product such that is a monoid and then V is a linear algebra over R or C or [22]. Definition 2 ([
22])
. Let V be a vector space over R (or C or ). A subspace of V is a subset W of V which is itself a vector space over R (or C or ) with the operations of addition and scalar multiplication as in V. Definition 3. Let V be a vector space over R (or C or ). A subset B of V is said to be linearly dependent or simply dependent if there exist distinct vectors, and scalars or C or not all of which are zero such that . A set which is not linearly dependent is called independent or linearly independent. If B contains only finitely many vectors we sometimes say are dependent instead of saying B is dependent.
The following facts are true [
22].
A subset of a linearly independent set is linearly independent.
Any set which contains a linearly dependent subset is linearly dependent.
Any set which contains the zero vector (0 vector) is linearly dependent for 1.0 = 0.
A set B is linearly independent if and only if each finite subset of B is linearly independent; that is if and only if there exist distinct vectors of B such that implies each .
For a vector space V over a field R or C or , the basis for V is a linearly independent set of vectors in V which spans the space V. We say the vector space V over R or C or is a direct sum of subspaces if and only if and is the zero vector for and .
The other properties of vector spaces are given in book [
22].
Now we proceed on to recall some essential definitions and properties of Neutrosophic Quadruples [
26].
Definition 4 ([
26])
. The quadruple where or C or , with as in classical Neutrosophic logic with a the known part and defined as the unknown part, denoted by or C or in called the Neutrosophic set of quadruple numbers. The following operations are defined on NQ, for more refer [
26].
For
and
in
[
26] have defined
are in NQ. For
in NQ and
s in
R or
C or
where
s is a scalar and
x is a vector in
V.
.
If in V usually termed as zero Neutrosophic Quadruple vector and for any scalar s in R or C or we have
Further for all or C or and . which is in NQ.
The main results proved in [
26] and which is used in this paper are mentioned below;
Theorem 1 ([
26])
. is an abelian group. Theorem 2 ([
26])
. is a monoid which is commutative. We mainly use only these two results in this paper, for more literature about Neutrosophic Quadruples refer [
26].
3. Neutrosophic Quadruple Vector Spaces and Their Properties
In this section we proceed on to define for the first time the new notion of Neutrosophic Quadruple vector spaces (NQ -vector spaces) their NQ vector subspaces, NQ bases and direct sum of NQ vector subspaces. All these NQ vector spaces are defined over R, the field of reals or C, the field of complex numbers and finite field of characteristic p, , p a prime. All these three NQ vector spaces are different in their properties and we prove all three NQ vector spaces defined over R or C or are of dimension 4.
We mostly use the notations from [
26]. They have proved
or
C or
,
p a prime;
is an infinite abelian group under addition.
We prove the following theorem.
Theorem 3. or C or ; p a prime, be the Neutrosophic quadruple group. Then is a Neutrosophic Quadruple vector space (NQ-vector space) over R or C or , where ‘∘’ is the special type of operation between V and R (or C or ) defined as scalar multiplication.
Proof. To prove
V is a Neutrosophic quadruple vector space over
R (or
C or
,
p is a prime), we have to show all the conditions given in Section two (Definition 1) of this paper is satisfied. In the first place we have
R or
C or
are field of scalars, and elements of
V we call as vectors. It has been proved by [
26] that
is an additive abelian group, which is the basic property on
V to be a vector space. Further the quadruple is defined using
R or
C or
,
p a prime, or used in the mutually exclusive sense. Now we see if
is in
V and
(or
C or
) then the scalar multiplication ‘∘’ which associates with each scalar
and the NQ vector
,
which is in
V, called the product of
n with
x in such a way that
for all or C or and .
is the zero vector of V and for 0 in R or C or ; we have .
Clearly is a vector space known as the NQ vector space over R or C or . □
However we can as in case of vector spaces say in case of NQ-vector spaces also is a NQ vector space with special scalar multiplication ∘.
We now proceed on to define the concept of linear dependence, linear independence and basis of NQ vector spaces.
Definition 5. Let be a NQ vector space over R (or C or ). A subset L of V is said to be NQ linearly dependent or simply dependent, if there exists distinct vectors and scalars (or C or ) not all zero such that . We say the set of vectors is NQ linearly independent if it is not NQ linearly dependent.
We provide an example of this situation.
Example 4. Let vector space over R. Let and be in V. We see so and z are NQ linearly dependent. Let and be in V. We cannot find a such that . If possible ; this implies , forcing , forcing ; forcing and forcing . Thus the equations are consistent and . So x and y are NQ linearly independent over R.
The following properties are true in case of all vector spaces hence true in case of NQ vector spaces also.
A subset of a NQ linearly independent set is NQ linearly independent.
A set L of vectors in NQ is linearly independent if and only if for any distinct vectors of L; implies each , for
We now proceed on to define Neutrosophic Quadruple basis (NQ basis) for , Neutrosophic Quadruple vector space over R or C or (or used in the mutually exclusive sense).
Definition 6. Let vector space over R (or C or ). We say a subset L of V spans V if and only if every vector in V can be got as a linear combination of elements from L and scalars from R (or C or ). That is if are n elements in L; then , is the NQ linear combination of vectors of L; where are in R or C or and not all these scalars are zero.
The Neutrosophic Quadruple basis for is a set of vectors in V which spans V. We say a set of vectors B in V is a basis of V if B is a linearly independent set and spans V over R or C or .
We say V is finite dimensional if the number of elements in basic of V is a finite set; otherwise V is infinite dimensional.
Theorem 5. Let be the Neutrosophic Quadruple vector space over R (or C or ). V is a finite dimensional NQ vector space over R (or C or ) and dimension of these NQ vector spaces over R(or C or ) are always four.
Proof. Let (or C or ), , be the collection of all neutrosophic quadruples of the Neutrosophic Quadruple vector space over R (or C or ). To prove dimension of V over R is four it is sufficient to prove that V has four linearly independent vectors which can span V, which will prove the result. Take the set contained in V; to show B is independent and spans V it enough if we prove for any , v can be represented uniquely as a linear combination of elements from B and scalars from R (or C or ). Now for the scalars (or C or ). Hence we see the elements of V are uniquely represented as a linear combination of vectors using only B, further B is a set of linearly independent elements, hence B is a basis of V and B is finite, so V is finite dimensional over R (or C or ). As order of B is four, dimension of all NQ vector spaces V over R (or C or ) is four. Hence the theorem. □
We call the NQ basis B as the special standard NQ basis of V.
Definition 7. Let be a NQ vector space over R (or C or ). A subset W of V is said to be Neutrosophic Quadruple vector subspace of V if W itself is a Neutrosophic Quadruple vector space over R (or C or ).
We will illustrate this situation by examples.
Example 6. Let be a NQ vector space over R. is a subset of V which is a NQ vector subspace of V over R. is again a vector subspace of V and is different from W.
We observe that the only common element between W and U is the zero quadruple vector .
Further it is observed if we define the dot product or inner product on elements in V. For and , denoted as ; and is in V. If for some then we say x is orthogonal (or dual) with y and vice versa. In fact . We say two NQ vector subspaces W and U are orthogonal (or dual subspaces) if for every and for every ; , that is two NQ vector subspaces are orthogonal if and only if the dot product of every vector in W with every vector in U is the zero vector.
is the zero vector subspace of V. Every NQ vector subspace of V trivial or nontrivial is orthogonal with the zero vector subspace of V. V the NQ vector space is orthogonal with only the zero vector subspace of V, and with no other vector subspace of V. W orthogonal U = = and = ; we call the pair of NQ subspaces as orthogonal or dual NQ subspaces of V.
Definition 8. Let be a Neutrosophic Quadruple vector space over R (or C or ); be n distinct NQ vector subspaces of V. We say is a direct sum of NQ vector subspaces if and only if the following conditions are true;
- 1.
Every vector can be written in the form , where are in R (or C or ) not all zero with .
- 2.
for and true for all varying in the set .
First we record that in case of all NQ vector spaces over R (or C or ) we can have the value of n given in definition to be only four, we cannot have more than four as dimension of all NQ vector spaces are only four. Secondly the minimum of n can be two which is true in case of all vector spaces of any finite dimension. Finally we wish to prove not all NQ vector subspaces are orthogonal and there are only finitely many nontrivial NQ vector subspaces for any NQ vector space over R (or C or ).
We prove as theorem a few of the properties.
Theorem 7. Let be a NQ vector space over R (or C or ). V has only finite number of NQ vector subspaces.
Proof. We see in case of NQ vector spaces over R (or C or ) the dimension is four and the special standard NQ basis for V is . So any non trivial subspace of V can be of dimension less than four; so it can be 1 or 2 or 3. Clearly there are some vector subspaces of dimension one given by, , , , , , , , , , , , , , and . Some the two dimensional vector spaces are , ;
in fact there are 105 NQ vector subspaces of dimension two. Further there are 1365 NQ vector subspaces of dimension three. Thus there are 1485 non trivial NQ vector subspaces in any NQ vector space over R (or C or ). We have shown that there are four NQ vector subspaces of dimension three all of them are hyper subspaces of V, of course we are not enumerating other types of dimension three subspaces generated by vectors of the form or are spaces of dimension three which we do not take into account as hyper subspaces. □
We define the three dimensional NQ vector subspace generated only by
is defined as the special pseudo Singled Valued Neutrosophic hyper NQ vector subspace of
V [
22,
24].