All the exercices we're doing in my linear algebra class use simple vectors when we're working on a basis, but every exam has a number that use a basis using $2 \times 2$ matrices. We've never covered that in class or examples so I have no idea how to approach it.
Example: Let $V$ be the vectorial space of the $2 \times 2$ square matrices, with the following basis:
$$ \beta = \{ e_1 = \begin{pmatrix} 1 & 0\\ 0 & 0 \\ \end{pmatrix}, e_2 = \begin{pmatrix} 0 & 1\\ 0 & 0 \\ \end{pmatrix}, e_3 = \begin{pmatrix} 0 & 0\\ 1 & 0 \\ \end{pmatrix}, e_4 = \begin{pmatrix} 0 & 0\\ 0 & 1 \\ \end{pmatrix}\}$$
Let $M = \begin{pmatrix} 1 & 2\\ 3 & 4 \\ \end{pmatrix}$ and $T$ the linear transformation of $V$ defined by $T(A) = AM$ where $A ∈ V$
Find the matrix $[T]_{\beta\beta}$, representing $T$ in the basis $\beta$
Normally, if I was working with a basis made out of vectors instead of matrices, I would make an augmented matrix with the vectors and put in the original coordinates of the vector and find the new coordinates using the Gauss Jordan method. But I can't make a matrix with other matrices, can't I? I'm at a complete loss.
What I've been able to work out so far:
$$x = [M]_{\beta\beta}$$ $$ x = \alpha_1 \begin{pmatrix} 1 & 0\\ 0 & 0 \\ \end{pmatrix} + \alpha_2 \begin{pmatrix} 0 & 1\\ 0 & 0 \\ \end{pmatrix} + \alpha_3 \begin{pmatrix} 0 & 0\\ 1 & 0 \\ \end{pmatrix} + \alpha_4 \begin{pmatrix} 0 & 0\\ 0 & 1 \\ \end{pmatrix} + $$
$$ x = \begin{pmatrix} \alpha_1 & \alpha_2\\ \alpha_3 & \alpha_4 \\ \end{pmatrix}(e_1 + e_2 + e_3 + e_4) $$
Hint:
A matrix $$ A=\begin{bmatrix}a&b\\ c&d \end{bmatrix} $$ in terms of the basis $\beta$ is expressed as: $$ A=ae_1+be_2+ce_3+de_4 $$
and we can write this as a ''vector'' of the components: $$A=\begin{bmatrix}a\\b\\ c\\d \end{bmatrix}$$
With this notation we have: $$ T(A)=[T]_{\beta \beta}\begin{bmatrix}a\\b\\ c\\d \end{bmatrix}=AM=\begin{bmatrix}a&b\\ c&d \end{bmatrix}\begin{bmatrix}1&2\\ 3&4 \end{bmatrix}= \begin{bmatrix}a+3b&2a+4b\\ c+3d&2c+4d \end{bmatrix}= \begin{bmatrix}a+3b\\2a+4b\\ c+3d\\2c+4d \end{bmatrix} $$
So you can use the usual ''machinery of vectors'' that you know to find the matrix $[T]_{\beta \beta}$.
The matrix that represents the transformation $T$ in the basis $\beta$ is: $$ [T]_{\beta \beta}= \begin{bmatrix} 1&3&0&0\\ 2&4&0&0\\ 0&0&1&3\\ 0&0&2&4 \end{bmatrix} $$
as you can easily verify.