Define the dual of a vector space $V$. State and prove a formula for its dimension.

Let $V$ be the vector space of real polynomials of degree at most $n$. If $\left\{a_{0}, \ldots, a_{n}\right\}$ are distinct real numbers, prove that there are unique real numbers $\left\{\lambda_{0}, \ldots, \lambda_{n}\right\}$ with

$$\frac{d p}{d x}(0)=\sum_{j=0}^{n} \lambda_{j} p\left(a_{j}\right)$$

for every $p(x) \in V$.