I am working with $\mathbb{R}^{nm}$ for some $n,m\geq3$. What is the minimum number of random vectors I need for them to span $\mathbb{R}^{nm}$ (in terms of $n$ and $m$)?
I'm happy for this to involve any sampling distribution (whichever is more convenient), but Uniform would be preferred.
Just so that you can mark it as answered:
Isn't the probability that you pick a hyper coplanar vector 0, since until the end the hyperplanes in which you pick vectors have measure 0. Maybe a 3d example will illustrate it better. When you pick your first vector everything's fine, when you pick your second the probability that it is collinear is 0 since the line has measure 0 wrt to your volume pdf, then when you pick your third vector the probability is once more 0 for it to be coplanar with the others, yielding that you'll always get vectors that span the whole space.
Note that due to machine precision (since you're effectively sampling a discrete probability in a computer) you may get a degenerate basis with a very small probability if you implement this on a computer.