Find image and kernel of the quotient map $q: V \to V / U$, $q(x)=[x]$
$U$ is a subspace of a linear vector space $V$. The definition of a quotient function is below:
Let $U$ be a subspace of a linear vector space $V$. Consider an equivalence relation of $V$ defined as $$x \sim y \quad \text{if} \quad x-y \in U.$$ The quatient space $V/U$ is then defined as the set of equivalence classes $[x]$ for all $x \in V$. The quatient space is a linear space, with operations defined as $$[x] + [y] := [x+y],\quad a[x]:= [ax] \quad \text{for } x,y \in V,\space a \in \mathbb{R}.$$
It seems natural to me that the kernel of such map could be $Ker(q)=\mathcal{O}$. But is this the only one and how can I prove it is actually a kernel?
Same with the image. There should be some vector that generates the whole image of the quotient map.
Any help would be appreciated.
You have $\ker q=U$, since\begin{align}q(v)=0&\iff v+U=U\\&\iff v\in U.\end{align}And $q$ is surjective; therefore, $\operatorname{Im}q=V/U$.