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 M be an affine subset of V.

Q1 (already solved)
Prove that M+a is affine for every a \epsilon V and that, if 0 \epsilon M, then M is a subspace.

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

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

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

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

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

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

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