Linear transformation problem M2x2 to P2 Can you guys help me with this question?

Let  $T:{M_{2\times2}} \to {P_2}$ be defined by $$T\begin{pmatrix} a&b\\c&d \end{pmatrix}=(a+b-c-d)t^2+ (c+d)t+ (a+b). $$ Find the matrix of T with respect to the standard bases for ${M_{2\times 2}}$ and ${P_2}$.

The standard bases for ${P_2}$ is $$\left\{ {1,t,{t^2}} \right\}$$ and for ${M_{2x2}}$ is $$\left\{ \begin{pmatrix}
   1 & 0  \\ 
   0 & 0  
\end{pmatrix},
\begin{pmatrix}
   0 & 1  \\ 
   0 & 0  
\end{pmatrix},
\begin{pmatrix}
   0 & 0  \\
   1 & 0  
\end{pmatrix},
\begin{pmatrix}
   0 & 0 \\
   0 & 1  
\end{pmatrix} \right\}$$

The matrix of a linear transformation comes from expressing each of the basis elements for the domain in terms of basis elements for the range upon applying the transformation.
For example,
$$T \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix} = 1\cdot 1 + 0\cdot t + 1 \cdot t^2.$$
Thus the first column of the matrix for $T$ with respect to these bases will be 
 $$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix}.$$
Repeat for the three remaining basis elements of $M_{2 \times 2}$. 
Solution:

 $$Mat(T) = \begin{bmatrix} 1 & 1 &0 & 0  \\ 0 & 0 & 1 &1  \\ 1 & 1 & -1 & -1\end{bmatrix} $$

