I having trouble solving this problem:
For the change of basis matrix of this transformation, I plugged in each entry of the basis $\{t_2, t, 1\}$ in for $T(p(t))$ like so:
$T(t^2)$ = some $2\times 2$ matrix... find the weights $c_1 \ldots c_n$ using the basis for $M(R)$ and put the weights in a vector
$T(t) =$ same
$T(1) =$ same
Then I am not sure on how to continue. I know that $T(t^2)$ will output some $2\times 2$ matrix, but how do I actually solve something like $T(t^2)$?
My attempt:
$T(t^2) = p(t) = t^2$, and $p(0) = 0, p(1) = 1, 0 = 0$, and $p'(0) = 0$ so the $2\times 2$ would be:
$\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$
For the size, I am guessing it wants the $m \times n$ size, I know for sure this change of basis matrix will be a $4 \times 3$ matrix.