  We prove that the unitriangular automorphism group of a free group of rank
$n$ has a faithful representation by matrices over a field, or in other words,
it is a linear group, if and only if $n \leq 3.$ Thus, we have completed a
description of relatively free groups with linear the unitriangular
automorphism groups. This description was initiated by Erofeev and the author
in \cite{Erofeev}, where proper varieties of groups have been considered.
