<s>[INST] Find $T(a+bx+cx^2)$ Let the linear transformation $T: \mathbb P_2 \rightarrow M_{2\times2}$ 
Find $T(a+bx+cx^2)$
knowing that:
$$T(1+x)=\begin{bmatrix} 1&0\\0&0\end{bmatrix}$$
$$T(x+x^2)=\begin{bmatrix} 0&1\\1&0\end{bmatrix}$$ 
$$T(1+x^2)=\begin{bmatrix} 0&0\\0&1\end{bmatrix}$$ 
I believe that it can be solved by combinations of the known transformations. But I've been stuck with this problem for a while. 
 [/INST] HINT
To start with, solve the following system in terms of $a,b$ and $c$.
\begin{align*}
a+bx+cx^{2} = p(1+x) + q(x+x^{2}) + r(1+x^{2}) \Longleftrightarrow
\begin{cases}
a = p + r\\
b = p + q\\
c = q + r
\end{cases}
\end{align*}
Once you know $p,q$ and $r$, you get
\begin{align*}
T(a + bx + cx^{2}) = pT(1+x) + qT(x+x^{2}) + rT(1+x^{2})
\end{align*}
Can you take it from here?
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