  We consider the repair problem for Reed--Solomon (RS) codes, evaluated on an
$\mathbb{F}_q$-linear subspace $U\subseteq\mathbb{F}_{q^m}$ of dimension $d$,
where $q$ is a prime power, $m$ is a positive integer, and $\mathbb{F}_q$ is
the Galois field of size $q$. For the case of $q\geq 3$, we show the existence
of a linear repair scheme for the RS code of length $n=q^d$ and codimension
$q^s$, $s< d$, evaluated on $U$, in which each of the $n-1$ surviving nodes
transmits only $r$ symbols of $\mathbb{F}_q$, provided that $ms\geq d(m-r)$.
For the case of $q=2$, we prove a similar result, with some restrictions on the
evaluation linear subspace $U$. Our proof is based on a probabilistic argument,
however the result is not merely an existence result; the success probability
is fairly large (at least $1/3$) and there is a simple criterion for checking
the validity of the randomly chosen linear repair scheme. Our result extend the
construction of Dau--Milenkovich to the range $r<m-s$, for a wide range of
parameters.
