In polynomial interpolation:
I see some connection between:
The Vandermonde matrix, the monomial basis and the fact that 'the monomial basis is not a good basis because it's components are not very orthogonal'.
Now, I still don't really grasp sufficiently the reason why exactly a Vandermonde matrix is often ill-conditioned. Also, I don't feel I understand why an orthogonal basis in general leads to better conditioned problems, how self-evident it may look from a certain point of view.