How can I show that, given a symmetric bilinear form in a finite dimensional vector space $V$ $\phi : V\times V \rightarrow \mathbb{R}$ then the vectors of the basis of $\ker \phi$ (which I assume that are always contained in any orthoghonal basis with respect to $\phi$) are linearly independent to the other vectors in the basis?
The problem is this: If I want to find an orthogonal basis of $V$ with respect to $\phi$ and $\phi$ is semidefinite positive I can consider the vector of a basis of $\ker \phi$, complete the basis to a basis of $V$ and then use Gram-Schmidt for the vectors of the built basis that do not belong to $\ker\phi$. I can do that because all the isotropic vectors are surely contained in $\ker\phi$ therefore the process works and what I find are vector surely orthogonal to the vectors of $\ker \phi$ (that are orthogonal to any other vector of $V$).
But then to be sure that what I have found is a basis of $V$ I must prove that the vectors in it are linearly indipendent and here is the problem.
How can I show that the vectors of $\ker \phi$ are linearly indipendent from the others?
Thanks in advice