Let $V$ be a vector space over a field $\mathbb{F}$.
Let $U \subset V$ such that $0_V \in U$, where $0_V$ is the zero vector of $V$, and $-u \in U \;\forall\;u \in U$, where $-u$ is the additive inverse of $u$.
Is $U$ together with the same operations as $V$ always a vector space over $\mathbb{F}$?
Or are there circumstances where this is not the case?
The reason I am asking this is because I have a list of questions that ask whether a particular subset is a vector space over $\mathbb{F}$. To me, it seems like they all are, because they all match the properties of $U$ above.
How about the union of two lines?
Is that closed under the addition operation?