Suppose that $V$ is the set of complex polynomials of degree at most $n$ in the variable $x$. Find the dimension of $V$ as a complex vector space.

Define

$$e_{k}: V \rightarrow \mathbf{C} \quad \text { by } \quad e_{k}(\phi)=\frac{d^{k} \phi}{d x^{k}}(0)$$

Find a subset of $\left\{e_{k} \mid k \in \mathbf{N}\right\}$ that is a basis of the dual vector space $V^{*}$. Find the corresponding dual basis of $V$.

Define

$$D: V \rightarrow V \quad \text { by } \quad D(\phi)=\frac{d \phi}{d x} .$$

Write down the matrix of $D$ with respect to the basis of $V$ that you have just found, and the matrix of the map dual to $D$ with respect to the dual basis.