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

• Mar 11th 2012, 01:07 PM
juanma101285
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 $M$ be an affine subset of $V$.

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 $\lambda \epsilon \mathbb{R}$."