  We establish the representability of the general linear ${\mathbb
Z}_2^n$-group and use the restricted functor of points - whose test category is
the category of ${\mathbb Z}_2^n$-manifolds over a single topological point -
to define its smooth linear actions on ${\mathbb Z}_2^n$-graded vector spaces
and linear ${\mathbb Z}_2^n$-manifolds. Throughout the paper, particular
emphasis is placed on the full faithfulness and target category of the
restricted functor of points of a number of categories that we are using.
