  Vector space interference alignment (IA) is known to achieve high degrees of
freedom (DoF) with infinite independent channel extensions, but its performance
is largely unknown for a finite number of possibly dependent channel
extensions. In this paper, we consider a $K$-user $M_t \times M_r$ MIMO
interference channel (IC) with arbitrary number of channel extensions $T$ and
arbitrary channel diversity order $L$ (i.e., each channel matrix is a generic
linear combination of $L$ fixed basis matrices). We study the maximum DoF
achievable via vector space IA in the single beam case (i.e. each user sends
one data stream). We prove that the total number of users $K$ that can
communicate interference-free using linear transceivers is upper bounded by
$NL+N^2/4$, where $N = \min\{M_tT, M_rT \}$. An immediate consequence of this
upper bound is that for a SISO IC the DoF in the single beam case is no more
than $\min\left\{\sqrt{ 5K/4}, L + T/4\right\}$. When the channel extensions
are independent, i.e. $ L$ achieves the maximum $M_r M_t T $, we show that this
maximum DoF lies in $[M_r+M_t-1, M_r+M_t]$ regardless of $T$. Unlike the
well-studied constant MIMO IC case, the main difficulty is how to deal with a
hybrid system of equations (zero-forcing condition) and inequalities (full rank
condition). Our approach combines algebraic tools that deal with equations with
an induction analysis that indirectly considers the inequalities.
