Show that a linear map is well-defined

Question

Let $V,W$ be vector spaces over field $F$ and $f:V\rightarrow W$ be a linear mapping such that $U \subseteq \ker(f)$. Show that the mapping $\overline{f}:V/U \rightarrow W$ given by $\overline{f}(v+U)=f(v)$ is a well-defined linear mapping. Show that $\ker(\overline{f})$=can$(\ker(f))$ where can$(\ker(f))=\{$can$(x) \mid x \in \ker(f)\}$

It looks like I need to apply first isomorphism theorem but I have no idea how to do that.

Answer 1

For your first question: if $v+U = w+U$, then $v-w\in U\subseteq \mathrm{ker}(f)$ and so $f(v) = f(w)$. Hence $\bar{f}(v+U) = f(v) = f(w) = \bar{f}(w+U)$. Thus, $\bar{f}$ is well-defined.