  In the first part of this paper, we prove a theorem which is the $q$-analogue
of a generalized modular Ray-Chaudhuri-Wilson Theorem shown in [Alon, Babai,
Suzuki, J. Combin. Theory Series A, 1991]. It is also a generalization of the
main theorem in [Frankl and Graham, European J. Combin. 1985] under certain
circumstances.
  In the second part of this paper, we prove $q$-analogues of results on a
recent notion called \emph{fractional $L$-intersecting family} for families of
subspaces of a given vector space. We use the above theorem to obtain a general
upper bound to the cardinality of such families. We give an improvement to this
general upper bound in certain special cases.
