Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B which is embedded with a stronger structure S.
By a proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A.
These types of structures occur in our every day's life, that's why we study them in this book.
Thus, as a particular case, we investigate the theory of linear algebra and Smarandache linear algebra.
A Linear Algebra over a field F is a vector space V with an additional operation called multiplication of vectors which associates with each pair of vectors u, v in V a vector uv in V called product of u and v in such a way that
i. multiplication is associative: u(vw) = (uv)w
ii. c(uv) = (cu)v = u(cv) for all u, v, w in V and c in F.
The Smarandache k-vectorial space of type I is defined to be a k-vectorial space, (A, , x) such that a proper subset of A is a k-algebra (with respect with the same induced operations and another ‘x’ operation internal on A), where k is a commutative field.
The Smarandache vector space of type II is defined to be a module V defined over a Smarandache ring R such that V is a vector space over a proper subset k of R, where k is a field.
We observe, that the Smarandache linear algebra can be constructed only using the Smarandache vector space of type II.
The Smarandache linear algebra, is defined to be a Smarandache vector space of type II, on which there is an additional operation called product, such that for all a, b in V, ab in V.
In this book we analyze the Smarandache linear algebra, and we introduce several other concepts like the Smarandache semilinear algebra, Smarandache bilinear algebra and Smarandache anti-linear algebra. We indicate that Smarandache vector spaces of type II will be used in the study of neutrosophic logic and its applications to Markov chains and Leontief Economic models – both of these research topics have intense industrial applications. |