I have a set of matrices $B=\{A_1, A_2, A_3, A_4\}$ that form a basis in $M_{2\times 2}(\mathbb{R})$.
I want to find the coordinates of a matrix $Y_E$ given in the standard basis in this new basis.
I know that the formula for finding these coordinates is: $$ \begin{equation} Y_E = T_{E\rightarrow B}Y_B \end{equation} $$ and so to find $Y_B$ we can do the following: $$ \begin{equation} T_{E\rightarrow B}^{-1}Y_E = T_{B\rightarrow E}Y_E = Y_B \end{equation} $$ In $\mathbb{R}^n$, this is easy because the transformation matrix is a normal matrix. But for this case, the transformation matrix is a matrix of matrices, so how can I find $Y_B$?
Thanks in advance.