I have some problems to deal with polynomials and matrix as vectors and basis of vector spaces, so I propose this exercise to understand what I should do, if someone would help me. Than you to all.
In $\mathbb{R_3}[t]$, the vector space of polynomials $deg(p(t)) \le 3$, consider the linear application $T$, so defined:
$a_o+a_1t+a_2t^2+a_3t^3 \to \begin{pmatrix} a_o+a_1 & 2a_2 \\ a_2-a_3 & a_0+a_1+2a_3 \end{pmatrix}$
write the matrices associated to $T$: in the basis $\{1, t, t^2, t^3\}$ of $\mathbb{R_3}[t]$ and in the basis $M_1:$$\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$, $M_2:$$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$, $M_3:$$\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}$, $M_4:$$\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}$ of $M(2,2, \mathbb{R^3})$.
This is what I did:
$T(1)= 1(M_1)+0(M_2)+0(M_3)+1(M_4)$;
$T(t)= 1(M_1)+0(M_2)+0(M_3)+1(M_4)$;
$T(t^2)= 0(M_1)+2(M_2)+1(M_3)+0(M_4)$
$T(t^3)= 0(M_1)+0(M_2)-1(M_3)+2(M_4)$
$\begin{pmatrix} 1 & 1&0&0 \\ 0 & 0&2&0\\0&0&1&-1\\1&1&0&2 \end{pmatrix}$