Matrices Linear transformation 

Reminder. In order to find the matrix coefficients $M_{ij}$ of a linear transformation $T$ wrt a basis $e_1,...,e_n$ of the doman and a basis $b_1,...,b_m$ of the codomain you need to express $T(e_j)$ in the basis $b_1,..., b_m$. The coefficient of $b_i$ in this expansion is the matrix coefficient $M_{ij}$, so:
$$T(e_j)=\sum_i M_{ij}b_i$$

a)
$$T( \ (-1 ,1 )\ ) = (-2-1, -3+2) = (-3,-1) = -3\cdot(1,0)+(-1)\cdot(0,1) $$
So we read off: $M_{11}=-3, M_{21}=-1$. Further:
$$T(\ (1,1)\ ) = (2-1, 3+2) = (1,5) = 1\cdot(1,0)+5\cdot(0,1)$$
and we read off $M_{12}=1$, $M_{22}=5$.
In total:
$$M=\begin{pmatrix}-3 &1 \\ -1 &5\end{pmatrix}$$

b)
$$T(\ (-1,1)\ ) =(-3,-1) =1\cdot(-1,1)+(-2)\cdot(1,1)$$
So we read off: $M_{11}=1$, $M_{21}=-2$. Further:
$$T(\ (1,1)\ ) = (1,5) = 2\cdot (-1,1)+3\cdot(1,1)$$
and we read off $M_{12}=2$, $M_{22}=3$.
In total:
$$M=\begin{pmatrix}1 & 2\\ -2&3 \end{pmatrix}$$