My LA textbook by Strang say there is no $2\times2$ matrix representation for linear transpose transformation on a $2\times2$ matrix. However, I figured out a way with matrices (not quite matrix representation though): $$ A^\top = \begin{bmatrix} 1 & 0\\ 0 & 0\\ \end{bmatrix}A\begin{bmatrix} 1 & 0\\ 0 & 0\\ \end{bmatrix}+ \begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}A\begin{bmatrix} 0 & 1\\ 0 & 0\\ \end{bmatrix}+\begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix}A\begin{bmatrix} 0 & 0\\ 1 & 0\\ \end{bmatrix}+\begin{bmatrix} 0 & 0\\ 0 & 1\\ \end{bmatrix}A\begin{bmatrix} 0 & 0\\ 0 & 1\\ \end{bmatrix} $$
I am pretty sure I figured it out mostly thanks to Strang's suggestive problem-set. I would like to know more about this trick, would anyone be kind enough to explain to me if there is anything deeper? or where should I begin to look up?