Let $S$ be a set of $n$ vectors from $\mathbb{Q}^d$. For every $k=1,2,\dots,n$, define $$r_k = \min_{T\subset S, |T|=k} \mathrm{rank}(T),$$ where $\mathrm{rank}(T)$ is the rank of a matrix formed by the vectors from $T$.

**Question 1.** Is there an efficient way to compute numbers $r_k$ ? I've developed some iterative procedure computing $r_1, r_2, \dots$ in order, but it is still rather slow.

**Question 2. (updated)** How to characterize sets $S$ that possess an ordering of elements, say, $S=\{ v_1, v_2, \dots, v_n \}$, such that for every $k=1,2,\dots,n$,
$$\mathrm{rank}(\{v_1, v_2, \dots, v_k\}) = r_k.$$
Christian Remling gave an example of $S$ that does not have such an ordering.