Find matrix of $T:P_1\to P_1$, given $T(at+b)=(a+2b)t+(4a+3b)$ w.r.t standard basis for $P_1$. What if basis changes to $\{2, t/2\}$?
I have come across this question, and it is obvious, the matrix of $T$ will be $\begin{bmatrix}1 & 4\\ 2 & 3\end{bmatrix}$, however I don't know how to find the matrix when basis is changed to $\{2, t/2\}$, can anyone please help to explain? Thank you.
Find a change of basis Matrix $A$ from your standard basis to the new basis $\{2, t/2\}$, then your new transformation $T'=ATA^{-1}$