# Ill-condioned Linear System and Gaussian Elimination

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.