If $0\stackrel{f}\rightarrow V \stackrel{g}\rightarrow 0$ is an exact sequence, then $V = 0$.

Question

Show that if $0\stackrel{f}\rightarrow V \stackrel{g}\rightarrow 0$ is an exact sequence of vector spaces over a field $K$, then $V = 0$.

I know that $\ker(g) = \text{im}(f)$, but how can I get that $V=0$?

Answer 1

Well, $\ker g=V$, since $g(v)=0$ for all $v\in V$. On the other hand, $\text{im}f=0 $, since the image of the zero vector space is always zero. The result follows.