Section 3.4 Theory
In this section.
Subsection 3.4.1 Polynomial interpolation
We have seen that Vandermonde matrices naturally arise as coefficient matrices in attempting to solve polynomial interpolation problems. Our geometric fact that these problems always have solutions can be formulated as an algebraic property of these matrices.
Theorem 3.4.1. Consistency of Vandermonde matrices.
A linear system whose coefficient matrix is a Vandermonde matrix is always consistent as long as the number of equations is no more than the number of variables and the second column contains no repeat entries. In the case that the number of equations is equal to the number of variables, there is one unique solution to the system.
Corollary 3.4.2. Consistency of the polynomial interpolation problem.
Given \(n+1\) points in the plane with different \(x\)-values, there is one unique polynomial of degree \(n\) or less that passes through all of the points.