Assuming $B$ is a symmetic bilinear form over a vector $\mathcal V$, about the null space of $B$, I found two different definition:
(1) $v\in \mathcal V$ is the element of null space, if $B(v,w)=0$ for all $w\in \mathcal V$.
(2) $v\in \mathcal V$ is the element of null space, if $B(v,v)=0$.
I want to know whether the two definition are same ? In fact, I feel they are different. If so, I want to know whether the dimension of the two null spaces by different definition are same ?