Prove that if $u+U=w+W$ then $U=W$ (where $u$,$w$ are vecotrs and $U$,$W$ are subspaces)

Question

having a hard time with this one:

$V$ is a subspace over $F$,

$u$ and $w$ are vectors in $V$,

$U$ and $W$ are subspaces of $V$

prove that if $u+U = w+W$ then $U=W$

($u+U$ is defined as $\{u+v~:~v \in U\}$)

Thanks in advance for your help. I know that in order to show that $U=W$ I need to show that they are included in each other from each direction but I just don't know how to show that...

Answer 1

This is equivilent to $U = (w-u) + W$, then $U = W$.

Denote $w-u$ by $\alpha$, so if $U \ne W$, WLOG, asuume $U \not \subset W$then there exists $x \in U$, $x \not \in W$. So $2x \in U,2x \not \in W$

Then, due to $U = \alpha + W$, we have there exists $w_1,w_2 \in W$, such that $x = \alpha + w_1, 2x = \alpha + w_2$, so $x = w_2 - w_1 \in W$.

A contradiction!