A question related to exact sequences and dimension of vector spaces
If $\require{cancel}n=3$, you have a short exact sequence$$0\longrightarrow V_1\overset{\alpha}{\longrightarrow}V_2\overset\beta\longrightarrow V_3\longrightarrow0.\tag1$$And $\alpha(V_1)=\ker\beta$. So, you can write $V_2$ as $\alpha(V_1)\oplus W$ and $\beta|_W$ is injective. Since $(1)$ is exact, it must be surjective too. So\begin{align}-\dim V_1+\dim V_2-\dim V_3&=\cancel{-\dim V_1}+\cancel{\dim V_1}+\bcancel{\dim W}-\bcancel{\dim V_3}\\&=0.\end{align}Can you deal with the general case now?