I am trying to show that there is a constant $C$ such that the following holds: Suppose $V$ is a linear vector space of dimension $k$ of vectors of length $n$ such that each nonzero vector of $V$ has at least $\epsilon n$ nonzero entries, for $0<\epsilon<0.1$. Then the projection of $V$ on a random set of $m=Ck/\epsilon \log(k/\epsilon)$ coordinates is a vector space of dimension $d$ with probability at least $1-\epsilon$.
I think I'm supposed to use VC dimensions and $\epsilon$-nets. Using the following theorem, I was thinking about letting the set of nonzero entries be the range, and the projection on the set of $m$ coordinates be the $\epsilon$-net. 