Exact short sequence

Question

Given a subspace $V_1$ of $V$, then $$ \{0\} \xrightarrow{} V_1 \xrightarrow{i} V \xrightarrow{\pi_{V/ V_1}} V/V_1 \xrightarrow{} \{0\} $$ is a exact short sequence.

Q: Is this something that I have to prove? What exactly do I have to prove?

Answer 1

For a sequence $A \overset{f}{\rightarrow} B \overset{g}{\rightarrow}C$, being exact in B means that $\operatorname{Ker}(g)=\operatorname{Im}(f)$. So you have to prove that your sequence is exact in $V_1$, $V$ and $V/V_1$, i.e. that the map $i$ is injective, the projection $\pi_{V/V_1}$ is surjective and that the image of the latter is equal to the kernel of the former. It is quite obvious but if you have any doubt, just prove it!