I have problems with understanding the role of inclusion mappings and quotient mappings in an exact sequence:
Let $V$ be a vector space. Let $U\subseteq V$ be a subspace of $V$. Then, $0\rightarrow U\rightarrow V\rightarrow V/U \rightarrow 0$ is an exact sequence.
Why does the inclusion mapping ($U\rightarrow V$) imply an injection between $U$ and $V$? Furthermore, why does $\ker(V\rightarrow V/U)=\text{im}(U\rightarrow V)$? Finally, why does $\ker(V/U \rightarrow 0)=\text{im}(V\rightarrow V/U)$?