I'm trying to prove the following statement:
Let $(M,g)$ be a semi-Riemannian manifold. For $X,Y\in T_pM$, prove that if $\langle X,V\rangle=\langle Y,V\rangle$ for all vectors $V\in T_pM$, then $X=Y$.
My approach is the following: We know that the metric $g$ has to be symmetric. Hence, it can be diagonalized. Let the diagonal metric be $g'$. If the eigenvalues are all non-zero, then it is easy to see that $\langle X,V\rangle=\langle Y,V\rangle$ for all vectors $V\in T_pM$ $\implies$ $X=Y$.
However, what if one of the eigenvalues is $0$? As this is a semi-Riemannian manifold, this is possible.