Vector Spaces & Bases · Challenge problem

In $P_2$ let $B = \{1, x, x^2\}$ and let $L = \{L_0, L_1, L_2\}$ be the Lagrange basis for the nodes $0, 1, 2$: each $L_i$ has degree at most $2$, $L_i(i) = 1$, and $L_i(j) = 0$ for the other two nodes $j$. Find the sum of all nine entries of the change-of-basis matrix $P_{L\leftarrow B}$, whose columns are the $L$-coordinate vectors of $1$, $x$ and $x^2$.