Let $V\subset k^N$ is a finite affine variety (strictly speaking, affine algebraic set), not necessarily irreducible. Show that if $|V|=n$, then $k[V]$, as $k$-algebra, is isomorphic to $k^n$.
I know that $k[V]$ is isomorphic to $k[X]/{\cal{I}} (V)$, writing $V=\{v_1,...v_n\}$, then ${\cal{I}} (V)={\cal{I}} (\{v_1\})\cap {\cal{I}} (\{v_2\})\cap...\cap {\cal{I}} (\{v_n\})$. But I am stuck at showing ${\cal {I}}(V)$ is isomorphic to $k^{N-n}$.
Please help, thank you!