Let $V=P_n(F)$ (the vector space of polynomials with coeffficients in R of degree at most n), and let $c_0,c_1,...,c_n$ be distinct scalars in F. For any scalars $a_1,...,a_n$, deduce that there exists a unique polynomial q(x) of degree at most n such that$ q(c_i)=a_i$ for $0 \leq i \leq n$. In fact $q(x)=\sum_{i=1}^n a_i p_i(x)$.
The hint says for uniqueness, prove that ${p_0(x),p_1(x),...,p_n(x)}$ is a basis for $V=P_n(F)$. My question is how to prove the existence? And after that we need to use the hint right?
The same problem gave you the recipe of how to construct the desired polynomial, just put $$q(x) = \sum_{j=0}^n a_jp_j(x)$$ and we check that this satisfy $q(c_i) = a_i$. But this is easy, since $$q(c_i) = \sum_{j=0}^n a_jp_j(c_i) = \sum_{j=0}^n a_j \delta_{ij} = a_i.$$