This reduces to the problem of finding a set of linearly independent vectors with maximum cardinality. There are in general many such sets, but any of them is a solution if you pick $c$ large enough. Then the convex hull is an $n$-orthoplex (AKA cross-polytope). If you make $c$ big, it will include any set of points in the span of the set, including on particular $a_1,...,a_m$. Some googling reveals that an algorithm for finding such a set is here.
Or are you trying to produce a solution with minimum $c$? This is a much more interesting question, by which I mean that I don't know the answer (:-).
