Weakly Generated Vector Spaces

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 W X is the subspace of V weakly generated by X , then X X W ⊇ , and W X ⊇X if and only if W X ⊇ X . Also, if X ⊇Y are subsets of V , then W W X ⊇ Y . If X is a finite subset of V and 0⊇X , then X is linearly independent if and only if X ⊇{0} is weakly independent. Also, we proved that the subset X of V is weakly independent if and only if each elementW v∋ X can be written as a weak linear combination of X as the only form. Finally, interesting properties and corollaries are obtained for weakly independent subsets.