  Suppose $X$ is a multidimensional diffusion process. Assume that at time zero
the state of $X$ is fully observed, but at time $T>0$ only linear combinations
of its components are observed. That is, one only observes the vector $L X_T$
for a given matrix $L$. In this paper we show how samples from the conditioned
process can be generated. The main contribution of this paper is to prove that
guided proposals, introduced in Schauer et al. (2017), can be used in a unified
way for both uniformly and hypo-elliptic diffusions, also when $L$ is not the
identity matrix. This is illustrated by excellent performance in two
challenging cases: a partially observed twice integrated diffusion with
multiple wells and the partially observed FitzHugh-Nagumo model.
