Vandermonde Matrix, Polynomial Interpolation & Orthogonal Basis

[azay]

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.

Any insights?

 

[trambolin]

Here is a rapid insight,
Obtain, the vector below, at each cases
$\left[\begin{array}{cc}1 &0\\0 &1\end{array}\right]\left(\begin{array}{c}x\\y\end{array}\right)= \left(\begin{array}{c}3\\5\end{array}\right)$


$\left[\begin{array}{cc}1 &-1\\1 &1\end{array}\right]\left(\begin{array}{c}x\\y\end{array}\right)= \left(\begin{array}{c}3\\5\end{array}\right)$

Please repeat it for the vector $$\left(\begin{array}{c}3.1\\5\end{array}\right)$$

You can relate orthogonality to being able to isolate arbitrary effects to one subset of orthogonal elements. Usually, complicated things are just for simplicity...