Find the matrix representation of the following linear transformation $T: M_{2x2} (R) \rightarrow P_{2} (R)$ defined by $T \begin{pmatrix} a & b \\ c & d \end{pmatrix}= (a+b) + (2d)x + bx^2$.
Let $\beta = \left \{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \right \}$ and $\gamma = {1, x, x^2}$. Compute $[T]_{\beta}^{\gamma}$.
This is what I have:
\begin{align}
[T]_{\beta}^{\gamma} &= \left [ [T\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}]_{\gamma} \, \, \, [T\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}]_{\gamma} \, \, \, [T\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}]_{\gamma}\, \, \, [T\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}]_{\gamma}\right ] 
&= \begin{pmatrix} 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 2x \\ 0 & x^2 & 0 & 0 \end{pmatrix}
\end{align}
but I'm not sure if it is correct. How can a matrix turn a 2x2 matrix into a polynomial?

Let $V$ a vector space and $\vec{v}$. The $\vec{v}$ representation with respect to a basis $B=\left\langle { \vec{\beta}_1, \cdots, \vec{\beta}_n } \right\rangle$ of $V$ is a colmun vector $$Rep_B(\vec{v})=\left( {\begin{array}{*{20}{c}}
   {c_1}  \\
    \vdots   \\
   {c_n}  \\
\end{array}} \right)
$$
such that $\vec{v}=c_1\vec{\beta}_1 + \cdots + c_n\vec{\beta}_n$. Now for to find the transformation matrix you need to find $Rep_B(T(\left( {\begin{array}{*{20}{c}}
   {1} & {0}  \\
   {0} & {0}  \\
\end{array}} \right)
))$, 
$Rep_B(T(\left( {\begin{array}{*{20}{c}}
   {0} & {1}  \\
   {0} & {0}  \\
\end{array}} \right)
))$, 
$Rep_B(T(\left( {\begin{array}{*{20}{c}}
   {0} & {0}  \\
   {1} & {0}  \\
\end{array}} \right)
))$ and $Rep_B(T(\left( {\begin{array}{*{20}{c}}
   {0} & {0}  \\
   {0} & {1}  \\
\end{array}} \right)
))$, where $B=\left\langle 1,x,x^2\right\rangle$, then put these four column vectors to form a matrix. According wiht this, in your matrix you will not have $x^2$ nor $2x$, but you will have $1$ and $2$, i.e. the respective coefficients, do you see why?.
Note that the matrix formed above will not transform from $2\times 2$ matrices to polynomials, it will transform from the representation of $2\times 2$ matrices to representation of polynomials with respect to given bases.
