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Suppose that I have a linear system $Ax=b$ such that $A$ is ill-conditioned. Can I say that it is dangerous to find a solution with Gaussian Elimination for this system, or does there exist some class of pivoting that allows me to find a good solution?
Does the Vandermonde Matrix have something special to do with this issue?
Ill conditioning is a property of the system of equations rather than the algorithm used to solve the system of equations. Using a bad algorithm can certainly make the situation worse, but you're already in trouble when you try to solve an ill-conditioned system of equations with A coefficients or right-hand side b with even tiny errors even if you use exact rational arithmetic. Similarly, you'll be in trouble if you use floating-point arithmetic with limited precision compared to the condition number.
Vandermonde matrices are typically extremely badly conditioned, so they are often used as examples of ill-conditioning.
