  We construct a new family of linearizations of rational matrices $R(\lambda)$
written in the general form $R(\lambda)=
D(\lambda)+C(\lambda)A(\lambda)^{-1}B(\lambda)$, where $D(\lambda)$,
$C(\lambda)$, $B(\lambda)$ and $A(\lambda)$ are polynomial matrices. Such
representation always exists and are not unique. The new linearizations are
constructed from linearizations of the polynomial matrices $D(\lambda)$ and
$A(\lambda)$, where each of them can be represented in terms of any polynomial
basis. In addition, we show how to recover eigenvectors, when $R(\lambda)$ is
regular, and minimal bases and minimal indices, when $R(\lambda)$ is singular,
from those of their linearizations in this family.
