# Finding a coset

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...

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What does $L$ have to do with the question? – Gerry Myerson Sep 25 '12 at 3:54
Anyway, you can "shift the zero vector" to an arbitrary element of $X$. – Gerry Myerson Sep 25 '12 at 3:55
@GerryMyerson Sorry, L is the "line" I'm referring to in defining X. – AsinglePANCAKE Sep 25 '12 at 3:56
The subspace is spanned by the differences between points of $X$ (i.e. vectors of the form $x-y$ with $x,y\in X$); the coset representative can be (as Gerry says) any element of $X$. It is helpful for these sorts of exercises to have a visual picture of what's going on, say with $V=\Bbb R^3$. Pick an arbitrary $u\in X$ and show that $U:=-u+X=\{x-u:x\in X\}$ is a vector subspace of $V$, and hence $X=u+U$ is a coset. – anon Sep 25 '12 at 4:34

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 $V/W$.