Weakly generated vector spaces

Abstract EN

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 0X , 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 element w vx 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.