Determinant criterion for a Chebyshev system ($\det (u_j(x_i))\ne 0$)

A family of $n+1$ real continuous functions is a Chebyshev system exactly when every evaluation matrix at $n+1$ distinct points is invertible. A singular evaluation matrix supplies a nonzero coefficient vector whose function vanishes at those points; conversely, a violation of the zero bound supplies such a vector in the kernel. This also gives the existence and uniqueness of interpolation from its span.