Find the matricial representation Let $M_{2}\mathbb{R}$ be the matrix space of $2 \times 2$ with real entries , and $P_{2}[x]$ the space of polinomies with real coeficientes with grade at most 2. Considerate the linear transformation $T: M_{2}(\mathbb{R}) \rightarrow P_{2}[x]$ defined like
$$T \begin{pmatrix} a & b \\ c & d \end{pmatrix}= (a+b)+2dx+bx^{2}$$
Calculate the matricial representation of $T$ respect the canonic base of $M_{2}(\mathbb{R})$ and $P_{2}[x]$ respectively:
$$B_{1}= \{ \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}, \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \} $$
$$B_{2}= \{1,x,x^{2} \} $$
I got $T(e_{1})=  1$, $T(e_{2})=1+x^{2}$, $T(e_{3})=0$, $T(e_{4})=2x$, so $$T= \begin{pmatrix} 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 2\\
0 & 1 &0 & 0
\end{pmatrix}$$
Am I right?

I think it is right:
$$\begin{pmatrix} 1 & 1 & 0 & 0 \\
0 & 0 & 0 & 2\\
0 & 1 &0 & 0
\end{pmatrix}\begin{pmatrix} a \\ b  \\  c   \\  d \\
\end{pmatrix}\,=\,\begin{pmatrix} a+b \\ 2d  \\  b \\
\end{pmatrix}$$
