  There is a large literature on cover-free families of finite sets, because of
their many applications in combinatorial group testing, cryptographic and
communications. This work studies the generalization of cover-free families
from sets to finite vector spaces. Let $V$ be an $n$-dimensional vector space
over the finite field $\mathbb{F}_{q}$ and let $\left[V\atop k\right]_q$ denote
the family of all $k$-dimensional subspaces of $V$. A family
$\mathcal{F}\subseteq \left[V\atop k\right]_q$ is called cover-free if there
are no three distinct subspaces $F_{0}, F_{1}, F_{2}\in \mathcal{F}$ such that
$F_{0}\leq (F_{0}\cap F_{1})+(F_{0}\cap F_{2})$. A family $\mathcal{H}\subseteq
\left[V\atop k\right]_q$ is called a $q$-Steiner system $S_{q}(t, k, n)$ if for
every $T\in \left[V\atop t\right]_q$, there is exactly one $H\in \mathcal{H}$
such that $T\leq H$. In this paper we investigate cover-free families in the
vector space $V$. Firstly, we determine the maximum size of a cover-free family
in $\left[V\atop k\right]_q$. Secondly, we characterize the structures of all
maximum cover-free families which are closely related to $q$-Steiner systems.
