Let $ 0 \to V_1 \to \dots \to V_n \to 0$ be an exact sequence of finite dimensional vector spaces over a field k. Prove that $\sum_{i=1}^n (-1)^i \dim_k V_i=0$.
Unfortunately, I have no intuition how should I proceed with this question. I know the properties of exact sequence but I am not able to make any progress. So, can you please give a hint or two on how should I approach the problem?
I have been doing a course in abstract algebra.
I would like to complete the question by myself.
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?