Odd dimension of a periodic Chebyshev system

A Chebyshev system of real continuous functions on a circle has odd dimension. If its dimension is $d$, take $d$ cyclically ordered evaluation points and move them continuously to the next cyclic permutation without letting points collide. The evaluation determinant stays nonzero, so cannot change sign. The final cyclic row permutation multiplies it by $(-1{)}^{d-1}$; consequently $d$ must be odd. For example, $1,\cos x,\sin x,\dots ,\cos mx,\sin mx$ has dimension $2m+1$ and the required zero bound.