Does a basis which is orthogonal with respect to a bilinear symmetric form $\phi$ necessarily contain the vectors of $ker \phi$ ?
Since the vectors in $ker \phi$ are orthogonal to all the vectors in the vector space $V$ where $\phi$ is defined, must they be present in all the orthogonal bases with respect to $\phi$?
Thanks for your help
Edit : with "vectors of $\ker \phi$" I mean the vectors of a basis of $ker \phi$
2nd Edit : $V$ is finite dimensional and $\phi : V \times V \rightarrow \mathbb{R}$ is defined for the reals