In this paper we consider linear relations with conjugates of a Salem number
$\alpha$. We show that every such a relation arises from a linear relation
between conjugates of the corresponding totally real algebraic integer
$\alpha+1/\alpha$. It is also shown that the smallest degree of a Salem number
with a nontrivial relation between its conjugates is $8$, whereas the smallest
length of a nontrivial linear relation between the conjugates of a Salem number
is $6$.