  Yang--Baxter maps (YB maps) are set-theoretical solutions to the quantum
Yang--Baxter equation. For a set $X=\Omega\times V$, where $V$ is a vector
space and $\Omega$ is regarded as a space of parameters, a linear parametric YB
map is a YB map $Y\colon X\times X\to X\times X$ such that $Y$ is linear with
respect to $V$ and one has $\pi Y=\pi$ for the projection $\pi\colon X\times
X\to\Omega\times\Omega$. These conditions are equivalent to certain nonlinear
algebraic relations for the components of $Y$. Such a map $Y$ may be nonlinear
with respect to parameters from $\Omega$.
  We present general results on such maps, including clarification of the
structure of the algebraic relations that define them and several
transformations which allow one to obtain new such maps from known ones. Also,
methods for constructing such maps are described. In particular, developing an
idea from [Konstantinou-Rizos S and Mikhailov A V 2013 J. Phys. A: Math. Theor.
46 425201], we demonstrate how to obtain linear parametric YB maps from
nonlinear Darboux transformations of some Lax operators using linear
approximations of matrix refactorisation problems corresponding to Darboux
matrices. New linear parametric YB maps with nonlinear dependence on parameters
are presented.
