Find the matrix $A$ of the linear transformation $T(f(t)) = 3 f'(t) + 6 f(t)$ from $P_2$ to $P_2$ with respect to the standard basis for $P_2$, $\left\lbrace 1, t, t^2 \right\rbrace$.

$A=$ $\left[\Rule{0pt}{3.6em}{0pt}\right.$$\left]\Rule{0pt}{3.6em}{0pt}\right.$