Im struggling to understand what bilinear symmetric forms and quadratic forms represent over a field, I understand the theory behind it, but I cant apply it.
For example I was given this exercise, which look simple and I get the intuition behind it, but I cant solve it concretely.
Find a non zero bilinear symmetric form such that its quadratic form associated is equal to $0$, for characteristic $2$
I think I got it, can someone verify my answer:
I define the bilinear symmetric form $f : \mathbb F_2^{2} \to \mathbb F_2$$:f((1,0),(1,0))=0 , f((1,0),(0,1))=1 , f((0,1),(0,1))=0$ which is non-zero and the matrix associated to the quadratic form obtained via $f$ is $$ \begin{pmatrix} 0 & 1 \\ 1 & 0 \\ \end{pmatrix} $$
which gives the quadratic form $2xy$ which is always equal to $0$ since we are in $\mathbb F_2$