Is showing that a subset of a vector space is spanned by a set of linearly independent vectors enough to conslude that the subset is a subspace?

Question

Sorry for any previous confusion. I think the title is asking what I want to ask now.

Essentially, the subset is described as having a specific form. If I show that any vector of that form is a linear combination of some vectors within that subset, is that subset a subspace?

I hope that was understandable!