  Given a linear system in P^n with assigned multiple general points we compute
the cohomology groups of its strict transforms via the blow-up of its linear
base locus. This leads us to give a new definition of expected dimension of a
linear system, which takes into account the contribution of the linear base
locus, and thus to introduce the notion of linear speciality. We investigate
such a notion giving sufficient conditions for a linear system to be linearly
non-special for arbitrary number of points, and necessary conditions for small
numbers of points.
