I'm dealing with an exercise on bilinear forms, but I don't know how to properly start. I have to prove that $\beta(A, B) = \mathrm{tr}(AB) - \mathrm{tr}(A)\mathrm{tr}(B)$ is a bilinear form, where $A, B$ are matrices and we are on the space $V = M_2(\mathbb{R})$.
Now, choosing two general matrices, I computed that $$\beta(A, B) = a_{12}b_{21} + a_{21}b_{12} - a_{11}b_{22} - a_{22}b_{11}$$
but I don't know if this was necessary.
According to the theory, a function $\phi: V\times V \to \mathbb{R}$ is a bilinear form on $V$ if it is linear in each variable separately.
I also computed the quadratic form $q: M_2(\mathbb{R}) \times M_2(\mathbb{R}) \to \mathbb{R}$, associated to that scalar product (id est, the bilinear form), which should be:
$$q(A, B) = 2a_{12}a_{21} - 2a_{11}a_{22}$$
Is that right?
Anyway to come back to the bilinear (symmetric) form, it's blurred to me how to start the proof. I know how to deal when I have two vectors like $\phi(v, w)$ but here I have two matrices. Or anyway, using the definition I calculated
$$\beta(A, B) = a_{12}b_{21} + a_{21}b_{12} - a_{11}b_{22} - a_{22}b_{11}$$
how should I proceed? It would be something like
$$\beta\Big((a_{11}, a_{12}, a_{21}, a_{22}), (b_{12}, b_{21}, b_{11}, b_{22})\Big)$$
and then I proceed applying the definition of bilinearity?
but this would make... well something like $\beta: \mathbb{R}^4\times \mathbb{R}^4 \to \mathbb{R}$....???