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A detailed walkthrough of Lagrange interpolating polynomials, covering how to find a polynomial that fits a set of data points exactly. Starts by framing the problem as a linear system using the Vandermonde matrix, then introduces Lagrange basis functions as a cleaner approach. Proves both existence and uniqueness of the interpolating polynomial via the Polynomial Interpolation Theorem, shows that Lagrange basis functions form a proper vector space basis for polynomials of degree at most n-1, and derives the identity matrix as the interpolation matrix in the Lagrange basis. Includes an appendix with an inductive proof of the Vandermonde determinant formula.

Notes on Lagrange Interpolating Polynomials

Lagrange polynomials as a basis for P_n(\mathbb{R})

Interpolation matrix in the Lagrange basis