Find the matrix representation $[T]_{\epsilon}^{\epsilon}$ with respect to $\epsilon$ I'm preparing for a preliminary graduate entrance exam, and i'm going over an old test. The question is worded exactly as follows, I am not leaving anything out. I am a bit confused as to what they are asking and (what is the matrix representation of $[T]_{\epsilon}^{\epsilon}$ with respect to $\epsilon$?), also, how to do it. Could somebody help me out? Thank you!
Consider the basis $\epsilon$ for $V=M_{2 \times 2}(\mathbb{R})$ with the following basis vectors:
$$ e_1 = \left( \begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix} \right) $$
$$ e_2 = \left( \begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix} \right) $$
$$ e_3 = \left( \begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix} \right) $$
$$ e_4 = \left( \begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix} \right) $$
And consider the linear transformation $T: V \rightarrow V$:
$T(\left( \begin{matrix} a & b \\ c & d \end{matrix} \right)) = \left( \begin{matrix} 2a-2b & -a+3b \\ 4c-2d & 3c-d \end{matrix} \right)  $
Find the matrix representation $[T]_{\epsilon}^{\epsilon}$ with respect to $\epsilon$
 A: $[T]_\epsilon^\epsilon$ means that the input and output vectors are in $\epsilon$ basis (here the standard basis).
Represent the $2\times2$ real matrix $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ by the $4\times1$ column vector $\begin{bmatrix}a\\b\\c\\d\end{bmatrix}$.
So we are given $T\begin{bmatrix}a\\b\\c\\d\end{bmatrix}=\begin{bmatrix}2a-2b\\-a+3b\\4c-2d\\3c-d\end{bmatrix}$.
Can you figure out the matrix of the linear transformation now?

 $M_T=\begin{bmatrix}2&-2&0&0\\-1&3&0&0\\0&0&4&-2\\0&0&3&-1\end{bmatrix}$

A: There's a good explanation on page 30 here: https://linear.axler.net/LinearAbridged.pdf (page 36 in the PDF, definition 3.32).
Suppose you are given finite dimensional vector spaces $V, W$, a linear map $T : V \to W$, and ordered bases $\alpha =\{v_1,\dots,v_n\}$ for $V$ and $\beta = \{w_1,\dots,w_m\}$ for $W$. Then the the matrix $[T]^\alpha_\beta$ is defined by
$$[T]^\alpha_\beta = \begin{pmatrix} [Tv_1]_\beta & \cdots & [Tv_n]_\beta \end{pmatrix}$$
where the columns are defined by
$$[x]_\beta = \begin{pmatrix} x_1 \\ \vdots \\ x_m \end{pmatrix} \iff x = \sum_{i=1}^m x_iw_i.$$
Thus to say the $(i,j)$-th entry of $[T]^\alpha_\beta$ is $A_{i,j}$ is to say
$$Tv_j = \sum_{i = 1}^m A_{i,j}w_i.$$
