Find the standard matrix representation of the linear transformation T in M2,2 let  $T: M_{2,2} \rightarrow M_{2,2}$ be a linear transformation defined by:
$$T \left(\begin{bmatrix}
        a & b\\
        c & d\\
        \end{bmatrix}\right) = \begin{bmatrix}a + b& b + a \\ c - d&d+b\end{bmatrix}
$$
Find the standard matrix for $S$ by using the standard basis of $M_{2,2}$
This is what i've done so far:
$$T \left(\begin{bmatrix}
        1 & 0\\
        0 & 0\\
        \end{bmatrix}\right) = \begin{bmatrix}1& 1 \\ 0&0\end{bmatrix} = x_1
$$
$$T \left(\begin{bmatrix}
        0 & 1\\
        0 & 0\\
        \end{bmatrix}\right) = \begin{bmatrix}1& 1 \\ 0&1\end{bmatrix} = x_2
$$
$$T \left(\begin{bmatrix}
        0 & 0\\
        1 & 0\\
        \end{bmatrix}\right) = \begin{bmatrix}0&0\\ 1&0\end{bmatrix} = x_3
$$
$$T \left(\begin{bmatrix}
        0 & 0\\
        0 & 1\\
        \end{bmatrix}\right) = \begin{bmatrix}0& 0\\ -1&1\end{bmatrix} = x_4
$$
But I'm not too sure where to go from here. All I know that these need to somehow become part of a larger matrix

Let $v_1,v_2,v_3,v_4$ denote the standard basis. Note that
$$
T(v_1) = T \left(\begin{bmatrix}
        1 & 0\\
        0 & 0\\
        \end{bmatrix}\right) = \begin{bmatrix}1& 1 \\ 0&0\end{bmatrix} = 
1v_1 + 1v_2 +0v_3 + 0v_4\\
T(v_2) = T \left(\begin{bmatrix}
        0 & 1\\
        0 & 0\\
        \end{bmatrix}\right) = \begin{bmatrix}1& 1 \\ 0&1\end{bmatrix} = 
1v_1 + 1v_2 + 0v_3 + 1v_4\\
T(v_3) = T \left(\begin{bmatrix}
        0 & 0\\
        1 & 0\\
        \end{bmatrix}\right) = \begin{bmatrix}0&0\\ 1&0\end{bmatrix} = 
0v_1 + 0v_2 + 1v_3 + 0v_4\\
T(v_4) = T \left(\begin{bmatrix}
        0 & 0\\
        0 & 1\\
        \end{bmatrix}\right) = \begin{bmatrix}0& 0\\ -1&1\end{bmatrix} = 
0v_1 + 0v_2 + (-1)v_3 + 1v_4
$$
It follows that the matrix of the transformation is given by
$$
S = 
\pmatrix{
1&1&0&0\\
1&1&0&0\\
0&0&1&-1\\
0&1&0&1
}
$$
