How to show a vector space is not closed under addition with elements not in the vector space.

Question

Wasn't entirely sure how to word the title. What I'm trying to show is:

Given $\vec{v}\in V$ and $\vec{w}\not\in V$, then $\vec{v}+\vec{w}\not\in V$

How would this statement be proven?

Answer 1

Hint: Suppose $v$ and $v+w$ were elements of $V$. Can you see which subspace properties allow us to deduce that $w \in V$?

Added Later: It is worth mentioning that the proper setting for this question is that $V$ is a subspace of some vector space $W$ so that it makes sense to add $v \in V$ and $w\notin V$; in particular, $w \in W\setminus V$. Then your question asks to show that $v + w \in W\setminus V$.