  Let $\text{PG}(n,q)$ be the Desarguesian projective space of dimension $n$
over the finite field of order $q$. The \emph{linear representation} of a point
set $\mathcal{K}$ in a hyperplane at infinity of $\text{PG}(n,q)$ is the
point-line geometry consisting of the affine points of $\text{PG}(n,q)$,
together with the union of the parallel classes of affine lines corresponding
to the points of $\mathcal{K}$. This type of point-line geometry has been
widely investigated in the literature. Curiously, if $\mathcal{K}$ is a
subgeometry, two disguises of its linear representation occur in two separate
works. In this short note, we give an explicit isomorphism between these two
disguises by making use of field reduction.
