Q:
If we interpolate n points by polynomials of degree n-1
then the interpolant is unique just in different basis. 
Given this, how can the conditioning for different forms of the interpolant 
be different, since they are they same polynomials, 
just represented in different basis?

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A:
The outcome is the same IF you process is exact.
When you need to invert a poorly conditioned matrix to compute the coefficients
then you may not get the exact interpolant.