Matrix representation wrt bases

Question

Consider the function $T:P_{2}\to P_{3}$ given by $T(p(x)) = xp(x)$. Let $\mathcal{B} = \{1, x, x^{2}\}$ be the ordered standard basis of $P_{2}$. Let $\mathcal{C} = \{2,x/2,3x^{2}, x^{3}-2\}$ be an ordered basis of $P_{3}$. Find the matrix representation of $T$ with respect to the bases $\mathcal{B}$ and $\mathcal{C}$. enter image description here

Answer 1

Hint: The columns will be $T$ applied to the elements of $\mathcal B$ (in order) expressed in terms of $\mathcal C$.