If $V$ is a $K$-vector space with $|K|\ge n$ and $V_1, \ldots, V_n$ are subspaces of $V$ such that $V=V_1 \cup \cdots \cup V_n$, then $V=V_i$ for some $i$. Is there a counterexample that for $|K|< n$ it doesn't hold ?
If $K$ has characteristic zero, it follows directly from induction that one of the subspaces has to be equal to $V$. Some hints are appreciated. Thanks