TITLE: When is the union of a family of subspaces of a vector space also a subspace?
QUESTION [5 upvotes]: It is not difficult to prove that the union of a chain (or, more generally, a directed family) of subspaces of a vector space $V$ is a subspace of $V$. 
Given a family $\mathcal{F}$ of subspaces of a vector space $V$ such that the union of $\mathcal{F}$ is a subspace of $V$, is it true that $\mathcal{F}$ is a directed family?
If not, is there a "nice" characterization of families of subspaces whose union is a subspace?

REPLY [4 votes]: The family $\mathcal{F}$ of one-dimensional subspaces of $V$ has the property that its union is the whole space, but it is not directed. Note that if $V$ is finite-dimensional over a finite field, then $\mathcal{F}$ is finite as well. I don't know of any characterisation though.