(i) Let $n \geqslant 1$ and let $x_{1}, \ldots, x_{n}$ be distinct points in $[-1,1]$. Show that there exist numbers $A_{1}, \ldots, A_{n}$ such that

$$\int_{-1}^{1} P(x) d x=\sum_{j=1}^{n} A_{j} P\left(x_{j}\right)$$

for every polynomial $P$ of degree $\leqslant n-1$.

(ii) Explain, without proof, how one can choose the points $x_{1}, \ldots, x_{n}$ and the numbers $A_{1}, \ldots, A_{n}$ such that $(*)$ holds for all polynomials $P$ of degree $\leqslant 2 n-1$.