Let $V$ be a $n$-dimensional vector space over $\mathbf{C}$ and let $(v_1^\ast,\ldots,v_n^\ast)$ be a basis for $V^\ast$. Then, there is a unique basis $(v_1,\ldots,v_n)$ for $V$ such that $v_i^\ast(v_j) =\delta_{ij}$.
What are some nice examples to illustrate this fact explicitly?
I know two examples.
Let $V$ be the vector space of polynomials of degree less than $n$. Let $a_1,\ldots,a_n$ be distinct complex numbers. Let $\textrm{ev}_{a_i}$ be the evaluation at $a_i$. Note that $(\textrm{ev}_{a_1},\ldots,\textrm{ev}_{a_n})$ is a basis for $V^\ast$. Thus, there are unique polynomials $P_1,\ldots,P_n$ of degree less than $n$ such that $P_i(a_j) =\delta_{ij}$. These are the Lagrange interpolation polynomials: $P_i(x) = \prod_{j\neq i} \frac{x-a_j}{a_j -a_i}.$
The second example is the standard example mentioned below in the comments section by Henning Makholm.
Are there any other nice examples?