Polynomial interpolation finds a polynomial function that passes exactly through a given set of distinct points. The Lagrange interpolation method provides a elegant solution using basis functions that are 1 at one point and 0 at all others. This approach guarantees both existence and uniqueness of the interpolating polynomial.

While the Vandermonde matrix approach works in theory, it suffers from numerical instability in practice. The Lagrange basis functions offer a more robust alternative, forming a linear algebra basis for the vector space of polynomials up to degree n-1. Each basis function is constructed to be 1 at its corresponding node and 0 at all other nodes.

The polynomial interpolation theorem states that for any set of n distinct points, there exists a unique polynomial of degree at most n-1 that interpolates them. This uniqueness is proven by showing that any two such polynomials would differ by a polynomial with more roots than its degree allows. The Lagrange basis provides a clean matrix representation where the interpolation matrix becomes the identity matrix, simplifying calculations significantly.