# Dual Space problems

Let $V=P_{n}(F)$ and let $c_{0},c_{1},c_{2},...,c_{n}$ be distinct scalars in F.
a. For $0 \leq i \leq n$, define $f_{i} \in V^{*}$ by $f_{i}(p(x)) = p(c_{i})$. Prove that $\{f_{0},f_{1},...,f_{n} \}$ is a basis for $V^*$
b. Show that there exist unique polynomials $p_{0}(x),p_{1}(x),...,p_{n}(x)$ such that $p_{i}(c_{j}) = \delta _{ij}$ for $0 \leq i \leq n$