Hi i have stumbled across this example of a quotient space but i am not quite sure i really understand what it means either algebraically or geometrically.
Fixing a vector space $V$ and a subspace $U$ of $V$. We can think of $V$ as the union of all the translates of this subspace $U$.
I know that by a ”translate” we mean a subset of the form
$v + U$ = {$v+u:u∈U$} = {$w∈V:w=v+u ∃u∈U$}
From there, i am trying to make sense of the following example:
Take $V= \mathbb{R}^2$ and $U$ to be the “x-axis”(the line with equation $y=0$).
Then for $v=(α_1,α_2)$,
$v+U$={$(α_1,α_2)+(α,0):α∈\mathbb{R}$} = {$(x,α_2):x∈\mathbb{R}$}. $(2)$
It is not clear to me why $U$={$α,0$} and not just {$x,0$} since $y=0$ and the "$x$" element is just $x$.
So here $v+U$ = {$(x,α_2):x∈\mathbb{R}$} but we could've written $α_1+α$ instead of x right?