Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 75 / 1 / a / Solution

Assume $a<b$. A Chebyshev system of dimension $n+1$ is a family of real continuous functions $u_0,\dots ,u_n$ for which every linear combination $u={\sum }_{j=0}^n{c}_j{u}_j$ with coefficients not all zero has at most $n$ distinct zeros in the interval. In particular the functions are linearly independent: an identically zero nontrivial combination would violate that bound. Multiplicities are not counted in this definition.

For distinct points $x_0,\dots ,x_n$, form the evaluation matrix $A=(u_j(x_i){)}_{i,j=0}^n$. If its determinant vanished, a nonzero vector $c$ would satisfy $Ac=0$. Its associated combination would vanish at all $n+1$ points, contradicting the Chebyshev system property. Conversely, if the property failed, some nonzero coefficient vector would give a combination vanishing at $n+1$ distinct points. At those points $Ac=0$, so the evaluation matrix would be singular. We have proved the determinant criterion for a Chebyshev system:This is also the existence-and-uniqueness condition for interpolation from the span of the functions: any prescribed $n+1$ data values give a uniquely solvable linear system.