Let $T: M_{22} \rightarrow M_{22}$ be defined by:
$$T \begin{pmatrix} \begin{bmatrix} a & b \\ c & d \end{bmatrix}\end{pmatrix} = \begin{bmatrix} 2c & a+c \\ b-2c & d \end{bmatrix}, $$ $B$ be the standard basis for $M_{22}$,
and
$$B' = \left \{ \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 1 & 1 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \right \} $$
Find the matrix $[T]_B$ of the linear transformation $T$ with respect to the basis $B$.
I'm having a hard time understanding exactly how to approach these questions. Any sort of direction would be very helpful.