How to show that [v] = v + U?

Question

There is given vector space $V$ through a field $K$ and $ U ⊂ V $ is a vector subspace.

There is the follwing mapping defined: $\phi : V/U \to V/U$.

As we know $V/U = \{ [v] \mid v \in V \}$ with $[v] = \{ w \in V \mid v - w \in U \}$. I could solve a problem if I could show that $[v] = v +U = \{ v+ u \mid u \in U \}= \{ w \in V \mid v - w \in U \}$ I know that it should be true, but I cannot prove it.

Let's say $ V = \mathbb{R}$ and $U = \{1,2,3,4,5\}$.

Let $v=7$

So it means, $[v] = \{ w \in V \mid v - w \in U \} =\{ 2,3,4,5,6\}$.

But $v+U = \{8,9,10,11,12 \}$.

Where am I missing the point? Can you please point it out?

Answer 1

Yes, your example simply cannot work because, as you state in your question it is a necessary condition that $U\subset V$ is a vector subspace. The set $\{ 1,2,3,4,5\}$ is not a vector subspace of $\mathbb{R}$ since it is not closed under addition, e.g. $3+5$ is not in $U$.

Try again with a better example, e.g. $V = \mathbb{R}^2$ and $U = \{ (x,y)\in \mathbb{R}^2| x=0\}$, and you will see how it works out well.

Answer 2

Let $A=\{w \in V: v-w \in U\}$ and $B=\{v+u:u \in U\}$. Then:

$w \in A \iff v-w=u$ for some $u \in U$ $ \iff w=v+(-u)$ for some $u \in U$ $\iff w \in B$