i've tried to solve this following question without any success. can you help me with it please?
let $V=M_{2 \times 2}^\mathbb{R}$ and $f: V \times V \to \mathbb{R}$ so that $f(A,B)=tr(A^tMB)$ for every $A,B \in V$. when n=2, $V=M_{2 \times 2}^{\mathbb{R}}$ and $M=\begin{pmatrix}1&2\\ 3&5\end{pmatrix}$
find a representation of f as a sum of a symmetrical bilinear form and an anti-symmetrical bilinear form.
what i tried:
since from the details we can infer that in order for f to be symmetric, M must be equal to $M^t$ ($M=M^t$). now for a bilinear form to be symmetrical we need that $B(u,v)=B(v,u)$ for every $u,v \in V$, and for a bilinear form to be anti symmetrical it must follow $B(u,v)=-B(v,u)$. but i don't get to finding a representation of f as a sum of a symmetrical and anti symmetrical bilinear forms using the given values.
please help me if you can.
thank you very much.
(edit: this is not a theoritical question about proving that every biliear map can be written as a sum of biliner symmetric map and a bilinear anti-symmetric map. This question is about implementing the theory using the given details in order to obtain the needed representation. additionally, on that thread the answer is incomplete and not enough for me in order to solve this question. thank you very much).