Let $ 0 \rightarrow V_1 \xrightarrow{\phi_1} V_2 \xrightarrow{\phi_2} V_3 \cdots \xrightarrow{\phi_n} V_n $ be an exact sequence of linear maps ($im(\phi_{i-1}) = ker (\phi_i)$ for all i. Show that if all $V_i$ are finite-dimensional then $\sum_{i=1}^{n} (-1)^n dim V_i = 0 $
I have being working on this problem and the way I understand it is that if I have a sequence of linear maps like.
$$ 0 \rightarrow V_1 \xrightarrow{\phi_1} V_2 \xrightarrow{\phi_2} V_3 \\ 0 \rightarrow v_{11} \rightarrow v_{21} \rightarrow 0 \\ 0 \rightarrow v_{12} \rightarrow v_{21} \rightarrow 0 \\ 0 \rightarrow v_{13} \rightarrow 0 \rightarrow 0 \\ 0 \rightarrow v_{14} \rightarrow v_{22} \rightarrow 0 \\0 \rightarrow v_{15} \rightarrow v_{22} \rightarrow 0 \\ \hspace{2cm} v_{23} \rightarrow v_{31} \\ \hspace{2cm} v_{24} \rightarrow v_{32} \\ \hspace{2cm} v_{25} \rightarrow v_{33} $$ I constucted this exact sequence of linear maps where $v_{1i}$ is the ith base vector for $V_1$ and $v_{2i}$ is the ith base vector for $V_2$ and so on. Yet $-5+5+3$ is not equal to $0$ what am I missing?
Thank you very much. I am note seeking a solution (at least not yet) I just want to understand why my reasoning is wrong.