Problem 5. Fix bases
$\mathcal{B} = (1, x, x^2)$, $\mathcal{C} = (1+3x+5x^2, 2+7x + 12x^2, 5 + 12x + 20x^2)$
of $P_2$.
(i) If $p(x)$ has coordinate vector $[7, 4, -3]^T$ with respect to $\mathcal{C}$, then what is its coordinate vector with respect to $\mathcal{B}$?
(ii) Find the coordinate vector of $5 + 8x + 13x^2$ with respect to $\mathcal{C}$.