Weakly Generated Vector Spaces

Abstract

It is important to appreciate at outset that the idea of a vector space in the algebraic abstraction and generalization of the Cartesian coordinate system introduced into the Euclidean plane, that is, a generalization of analytic geometry. Therefore, a number of interesting papers have been published on the concepts of generating sets and linearly independence. In this paper, we study the notion of weak generation of a vector space over a field and the notion of weakly independent sets as a generalization of linearly independent sets in vector spaces. We proved that if is the subspace of weakly generated by , then , and if and only if . Also, if  are subsets of , then . If  is a finite subset of  and , then  is linearly independent if and only if  is weakly independent. Also, we proved that the subset  of  is weakly independent if and only if each element  can be written as a weak linear combination of  as the only form. Finally, interesting properties and corollaries are obtained for weakly independent subsets.