I'm having trouble with the following problem:
Let $\tau_A: F^2\times F^2 \rightarrow F$ be a symmetric bilinear form given by $\tau_A (v,w)=v^tAw$, $\forall v,w\in F^2$ and $A=\begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}$, where $F$ is a field.
Suppose the characteristic of $F$ is not equal to $2$. Prove that there exists a basis $\mathcal{B}=\{v_1,v_2\}$ of $F^2$ such that $\tau_A (v_1,v_1)=\tau_A (v_2,v_2)=0$
So far I've tried seeing if I could milk anything out of the non-degeneracy of $\tau_A$ (so that $(F^2,\ \tau_A)$ is an inner product space), but got stuck. I also split this problem into two cases: $Char(F)=0$ and $Char(F)=p$, but wasn't able to get anywhere. I have no experience dealing with the characteristic of a field, so I think conceptually I'm having a hard time understanding why it would matter in a problem like this.
Any tips or solutions (preferably as elementary as possible) would be appreciated! Thanks in advance!