Proving that $S(A):=(A+A^t)/2$ is self-adjoint

Question

I am having quite a hard time solving the following question:

The standard inner space product is defined in the question.

All of the questions are correct, but I don't know how to prove it.

I know that for the first question, I need that a linear transformation be

$$ (Tu,v)=(u,Tv). $$

Therefore, I try to solve the question with inner product, but every time I reach the following result:

The way I get to the result is with the rules of inner product, such as:

$$(u+v,w)=(u,w)+(v,w),$$ and using matrices algebra instead of linear transformation.

Unless I can prove that $A$ and $A^t$ are equal (and so are the $B$ and $B^t$), I can't see how I can prove the question.

Your help is appreciated, thank you.

Answer 1

The "standard inner-space product" which you mention is $$ (A,B):=\mathrm{Tr}(A^tB),\quad A,B\in V. $$ Using this definition, do you see how to prove that $$ (A^t,B)=(A,B^t)? $$