Vandermonde matrix

Summary

A Vandermonde matrix has rows of the form $1, x, x^2, \dots, x^{n-1}$, one row for each of the numbers $x_1, \dots, x_m$. Multiplying it by the coefficient vector of a polynomial gives the values of the polynomial at $x_1, \dots, x_m$. Its determinant, the Vandermonde determinant, is nonzero exactly when the $x_i$ are distinct. This is why a polynomial of degree less than $n$ is fixed by its values at $n$ distinct points.