the KEY example of a dual basis:
Let $V=\cP_d$, polynomials of degree at most $d$,
``evaluation at $\alpha_0,\dots,\alpha_d$ (all not equal)''
  is a basis for $V^*$;
the -dual- basis for $\cP_d$ is
$p_i(\alpha_j)=\delta_{ij}$
Concretely, it's
$$p_i(x)=\prod_{j\neq i}(x-\alpha_j)/(\alpha_i-\alpha_j)$$

These are called the ``Lagrange interpolated polynomials''

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Why is this -extremely important-?
B/c it's a very physical, hands-on example of the abstract definition,
where you can actually work it out!
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A more pedestrian example are the Taylor polynomials,
which are dual to $p \mapsto p^{(i)}(0)$;
the basis is $x^i/i!$.
