How to show that $\{ w \in V \mid v - w \in U \} = \{ v+ u \mid u \in U \} $

Question

There is a vectorspace $V$ and its subvectorspace $ U$ : $(U ⊂ V)$

There is $[v] ∈ V/U$ which means $[v] = \{ w \in V \mid v - w \in U \}$.
How is it is possible to show, that $\{ w \in V \mid v - w \in U \} = \{ v+ u \mid u \in U \}$ ?

Answer 1

It is an elementary calculation. Define $A=\{w\in V~|~v-w\in U\}$ and $B=\{v+u~|~u\in U\}$. You have to prove $A=B$.

Hint 1: Prove $A\subseteq B$ and $B\subseteq A$.

Hint 2: What means $w\in A$ and $w\in B$?

Solution:

$A\subseteq B$:

Let be $w\in A$, then $v-w\in U$. This yields that there exists $u\in U$ such that $v-w=u$. Now you have $w=v-u=v+(-u)$ Since $U$ is a subvectorspace of $V$ you have $-u\in U$ and therefore $w\in B$.

and $B\subseteq A$:

Let be $w\in B$, then there exists $u\in U$, such that $w=v+u$. This yields $w-v=u\in U$ and therefore $w\in A$.