let $f: \Bbb R^2 \times \Bbb R^2 \to \Bbb R$ a symmetric bilinear form. s.t $\forall g\in M_2 (\Bbb R) \forall u,v\in \Bbb R^2 : f(gu,gv)=f(u,v)$ Find the possible values of $\det (g) $.
Thoughts I tried using the formula to transfer it to a quadratic form, but this didn't seem to yield much. Also thought of putting it like this $u^tgAgv=u^tAv$ but I don't know what to do next with this (taking the det on both sides retains the same equality).
That depends. Assuming your bilinear form is non-degenerate and its representation matrix is, say, $S$, then I get from what you've provided:
$f(gu,gv)=u^tg^tSgv = u^t S v = f(u,v)\quad \forall u,v\in\mathbb{R}^2$ therefore $g^t S g = S$
Assuming $f$ is non-degenerate then $S$ is regular and we get $ det(g^t S g)= det(g^t) det(S) det(g) = det(S)$ which would yield $det(g)^2 = 1\Rightarrow det(g)=\pm 1$