Find the matrix of the linear transformation \(T:\mathbb {M}_{2,2}\rightarrow \mathbb {M}_{2,2}\), given by \(T(A)=A^T\), if the basis of the domain (\(\mathcal {B}\)) and the basis of the codomain (\(\mathcal {D}\)) are given by
\[\mathcal {B}=\mathcal {D}=\left \{\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix},\begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\1 & 0\end{bmatrix}, \begin{bmatrix}0 & 0\\0 & 1\end{bmatrix}\right \} \]
\[\begin{bmatrix}\answer {1} & \answer {0} & \answer {0} & \answer {0}\\\answer {0} & \answer {0} & \answer {1} & \answer {0}\\\answer {0} & \answer {1} & \answer {0} & \answer {0}\\\answer {0} & \answer {0} & \answer {0} & \answer {1}\end{bmatrix}\]

Source

[Nicholson] W. Keith Nicholson, Linear Algebra with Applications, Lyryx 2021, Open Edition, Exercises 9.1.3 (b).