  We study the functor $\operatorname{Def}_E^k$ of infinitesimal deformations
of a locally free sheaf $E$ of $\mathcal{O}_X$-modules on a smooth variety $X$,
such that at least $k$ independent sections lift to the deformed sheaf, where
$h^0(E) \geq k$. We deduce some information on the $k$-th Brill-Noether locus
of $E$, such as the description of the tangent cone at some singular points, of
the tangent space at some smooth ones and some links between the smoothness of
the functor $\operatorname{Def}_E^k$ and the smoothness of some well know
deformations functors and their associated moduli spaces. As a tool for the
investigation of $\operatorname{Def}_E^k$, we study infinitesimal deformations
of the pairs $(E,U)$, where $U$ is a linear subspace of sections of $E$. We
generalise to the case where $E$ has any rank and $X$ any dimension many
classical results concerning the moduli space of coherent systems, like the
description of its tangent space and the link between its smoothness and the
injectivity of the Petri map.
