For a general polytope \(P\), a vertex of \(P\) is defined as an extreme point of \(P\), that is, it is not a convex combination of other points of \(P\).

(a) Show that for a simplex \(S\) given as the convex hull of the affinely independent points \(v^{1}, \ldots, v^{d}\), each point \(v^{i}\) is indeed a vertex of \(S\) according to this definition.

(b) Show that the set of vertices of a simplex is unique.