Let $L/K$ be a field extension. Consider two $n\times n$ matrices $A$ and $B$ over a field K. Suppose that there exist nonzero vectors in $L^n$ such that: $$Ax=0, By=0, x^T\cdot y =0,$$
Surely matrix equations has solutions in $K$.
Is it possible to find such vector $x',y'\in K^n$ that will also have zero product?