## Deducing that there exists a subspace U of V and a of V such that M=U+a

Hi, I have the following problem I have been stuck on for the last couple of hours ... Thanks for any help you can give me!

It says:

"Let $\displaystyle M$ be an affine subset of $\displaystyle V$.

Prove that $\displaystyle M+a$ is affine for every $\displaystyle a \epsilon V$ and that, if $\displaystyle 0 \epsilon M$, then $\displaystyle M$ is a subspace.

Q2 (the one I am stuck on)
Deduce that there exists a subspace $\displaystyle U$ of $\displaystyle V$ and $\displaystyle a \epsilon V$ such that

$\displaystyle M=U+a$ ----> this is formula (1)

[Hint: what can be said about such an $\displaystyle a$, assuming that it exists?]

Show further that the subspace $\displaystyle U$ in (1) is uniquely determined by $\displaystyle M$ and describe the extent to which $\displaystyle a$ is determined by $\displaystyle M$."

And other details regarding the problems in this part of the project are:

"Throughout Part A of this project, $\displaystyle V$ will be a real vector space and, for a non-empty subset $\displaystyle S$ of $\displaystyle V$ and $\displaystyle a \epsilon V$, the set {$\displaystyle x+a : x \epsilon S$} will be denoted by $\displaystyle S+a$.

An affine subset of $\displaystyle V$ is a non-empty subset $\displaystyle M$ of $\displaystyle V$ with the property that $\displaystyle \lambda *x+(1- \lambda )*y \epsilon M$ whenever $\displaystyle x,y \epsilon M$ and $\displaystyle \lambda \epsilon \mathbb{R}$."