About transpose matrix transformation problem. I have this problem that I don't understand so I can't solve. I wish someone could explain me it or solve it.

Let $M_2(\mathbb{R})$ the vector space generated by all the square
  matrices of $2\times 2$. Consider the linear transformation $T\colon
 M_2(\mathbb{R})\longrightarrow M_2(\mathbb{R})$ given by $T(A)=A^T$
  (where $A^T$ is the transpose of $A$). Calculate a basis for
  $M_2(\mathbb{R})$ such that the transformation $T$ is represented by a
  diagonal matrix. Which are the possible values for the diagonal?

 A: Let me just explain what the problem is asking you to do since this seems to be what you are wanting to know. 
The point is that all linear transformations can be represented by a matrix. So you want to find a matrix $A$ such that $T(v) = Av$. Now, this is not hard, you just need to determine what $T$ does to basis elements of $M_2(\mathbb{R})$. Now you need to actually pick a basis such that $A$ is a diagonal matrix.
Let's just try. Let
$$
A = \pmatrix{1 & 0 \\ 0 & 0}, B = \pmatrix{0 & 1 \\ 0 & 0}, C = \pmatrix{0 & 0 \\ 1 & 0}, D = \pmatrix{0 & 0 \\ 0 & 1}.
$$
Then you have a basis for the 4 dimensional vector space. So a matrix for $T$ will be a $4\times 4$ matrix.
Now
$$
T(A) = 1A + 0B + 0C + 0D \\
T(B) = 0A + 0B + 1C + 0 D \\
$$
and so on. So a matrix for $T$ relative to this basis would look something like this
$$
\pmatrix{
1 & 0 & ? & ? \\
0 & 0 & ? & ? \\
0 & 1 & ? & ? \\
0 & 0 & ? & ?
}
$$
So that is the point. Now try to find another basis so that you get a diagonal matrix here.
