Consider a Chebyshev system $g_0,...,g_n \in C[a,b]$ and $(n+1)$ value pairs where $x_i\neq x_j$ for $i\neq j$ that are all in $[a,b]$. Prove that there is exactly one element $g \in span(g_0,...,g_n)$ where the interpolation condition $g(x_i) = y_i, i = 0...n$ is fulfilled.
My thoughts: Well first, I have to prove that such an element exists, then I need to show that it is unique. I know that in a Chebyshev system every non-trivial linear combination has no more than n roots in [a,b]. How can I use this information or how should I start?