Lemma:

If $w(f,g) \equiv 0$ ( $w(f,g)(x)=0, \forall x$), then $f,g$ are linearally dependent. If $w(f,g) \not\equiv 0$ ($\exists x$ s.t. $w(f,g)(x) \neq 0$), then $f,g$ are linearally independent.