I'm given $V$ a vector space over a field $\mathbb{F}$. Letting $v_1$ and $v_2$ be distinct elements of $V$, define the set $L\subseteq V$: $L=\{rv_1+sv_2 | r,s\in \mathbb{F}, r+s=1\}$. (This is the line through $v_1$ and $v_2$.
And denote by $X$ a non-empty subset of $V$ which contains all lines through two distinct elements of $X$.
I'm supposed to show that $X$ is the coset of some subspace of $V$. This is a bit of a follow up to this, where I was too quick to assert that understanding what $X$ is would make solving the problem easy.
I'm just not sure for an arbitrary $X$ where to shift the zero vector...
For every line $L \in X$, associate a vector $v$ that is the difference of two distinct points on the line. Let $W$ be the subspace of $V$ generated by all these vectors. Show that $X$ is a coset of $V/W$.