# New Classes of Neutrosophic Linear Algebras by W. B. Vasantha Kandasamy

By W. B. Vasantha Kandasamy

During this booklet we introduce 3 sorts of neutrosophic linear algebras: neutrosophic set lineat algebra, neutrosophic semigroup linear algebra, and neutrosophic team linear algebra. those are generalizations of neutrosophic linear algebra. those new algebraic constructions pave the best way for purposes in numerous fields like mathematical modeling.

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If V in addition being a set neutrosophic modulo integer vector space over S satisfies the condition, that for every pair v, u  V, u + v and v + u  V; then we call V to be a set neutrosophic modulo integer linear algebra over S. We illustrate this by some simple examples. 23: Let V = {0, I, 2I, 3I, 4I, 5I, 6I, 7I, 8I, 9I, 10I}  N(Z11) and S = {0, 1, I, 5, 3, 2I, 6I, 8I}  N(Z11). V is a set neutrosophic modulo integer linear algebra over S. 24: Let V = {0, 2I, 4I, 6I, 8I, 10I, 12I, 14I, 16I}  N(Z18), S = {0, 1, 2, 4, 8, 2I, 6I, 10I}.

We say V is a neutrosophic-neutrosophic integer set vector space over S  N(Z) (S  Z) if sQi = Qi s  V for every Qi  V and s  S. We shall for easy representation write neutrosophicneutrosophic integer vector space as n-n integer set vector space. We now illustrate this new structure by some examples. 1: Let V = {0, 1 + (2n – 1)I | n = 1, 2, …, f}  N(Z). V is a n-n integer set vector space over S = {0, 1 + I, 1}  N(Z). 2: Let V = {I, 2I, 5I, 7I, 0, 8I, 27I}  N(Z). V is a n-n integer set vector space over S = {0, I}  N(Z).

So PN(Q) hereafter will be known as pure set neutrosophic rational numbers. 1: Let V  N(Q) (PN(Q)) be a proper subset of N(Q) or V contains elements from N(Q) (PN(Q)) (V  Q). Let S  N(Q) be a proper subset of N(Q). We say V is a mixed (pure) set neutrosophic rational vector space over S if sQ  V for every s  S and Q  V. 1: Let 19 19I 27 27I ­ 2 2I  ,  , V = ®  , 0, 2 2 5 5 ¯7 7 17I, 48 – 48I, 28 28I 47I ½  , ¾  N(Q). 13 13 5 ¿ Take S = {0, 1, 11 11I , 1 – I}  N(Q).  7 7 It is easily verified V is a pure set neutrosophic rational vector space over the set S.