Let $\displaystyle V=P_{n}(F)$ and let $\displaystyle c_{0},c_{1},c_{2},...,c_{n} $ be distinct scalars in F.

a. For $\displaystyle 0 \leq i \leq n$, define $\displaystyle f_{i} \in V^{*} $ by $\displaystyle f_{i}(p(x)) = p(c_{i}) $. Prove that $\displaystyle \{f_{0},f_{1},...,f_{n} \} $ is a basis for $\displaystyle V^*$

b. Show that there exist unique polynomials $\displaystyle p_{0}(x),p_{1}(x),...,p_{n}(x)$ such that $\displaystyle p_{i}(c_{j}) = \delta _{ij} $ for $\displaystyle 0 \leq i \leq n $

I'm really lost in these problems, perhaps I don't have a good understanding of dual space, please help!