Let $K$ be a field and $V = M_{n, n}(K)$ the ring of $n \times n$ matrices over $K$. For any $f \in V^*$ (the dual space of $V$), we set:
$\gamma_f: V \times V \to K, (A, B) \mapsto f(A B^t)$.
I now first want to show that $\gamma_f$ is a bilinear form. Furthermore, I want to prove that:
$\Gamma: V^* \to Bil(V), f \mapsto \gamma_f$
is linear and injective. (Where ($Bil(V)$ is the set of bilinear forms from $V \times V$ to $K$.)
Thanks in advance. Bilinear forms are new to me, especially when it comes to working with them in regards to matrices.