> 
> Let $V$ be a $\mathbb C$-vector space, $\dim V>1$. Then for any bilinear form $\phi$ there is $v\in V$ s.t. $\phi(v,v)=0$ (so there is no positiv definite bilinear form on $V$)
> 
> 
> 

If one takes the usual multiplication then $\begin{pmatrix}i\\0\end{pmatrix}\cdot \begin{pmatrix}i\\0\end{pmatrix}=-1$, so for a particular bilinear form, positive definiteness fails, but I need some example which gives $0$ and for any bilinear form.