= Solution
Assume $a<b$. A <Chebyshev system> of dimension $n+1$ is a family of real <continuous functions> $u_0,\ldots,u_n$ for which every <linear combination> $u=\sum_{j=0}^n c_ju_j$ 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 $x_0,\ldots,x_n$, 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>:
$$
\boxed{\Phi\text{ is a Chebyshev system}\iff
\det(u_j(x_i))_{i,j=0}^n\ne0\text{ for every set of distinct }x_i.}
$$
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.
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