  Let $V$ be an $(n+\ell)$-dimensional vector space over a finite field, and
$W$ a fixed $\ell$-dimensional subspace of $V$. Write ${V\brack n,0}$ to be the
set of all $n$-dimensional subspaces $U$ of $V$ satisfying $\dim(U\cap W)=0$. A
family $\mathcal{F}\subseteq{V\brack n,0}$ is $t$-intersecting if $\dim(A\cap
B)\geq t$ for all $A,B\in\mathcal{F}$. A $t$-intersecting family
$\mathcal{F}\subseteq{V\brack n,0}$ is called non-trivial if
$\dim(\cap_{F\in\mathcal{F}}F)<t$. In this paper, we describe the structure of
non-trivial $t$-intersecting families of ${V\brack n,0}$ with large size. In
particular, we show the structure of the non-trivial $t$-intersecting families
with maximum size, which extends the Hilton-Milner Theorem for ${V\brack n,0}$.
