  Let $V$ denote an $r$-dimensional $\mathbb{F}_{q^n}$-vector space. Let $U$
and $W$ be $\mathbb{F}_q$-subspaces of $V$, $L_U$ and $L_W$ the projective
points of $\mathrm{PG}\,(V,q^n)$ defined by $U$ and $W$ respectively. We
address the problem when $L_W=L_U$ under the hypothesis that $U$ and $W$ have
maximum dimension, i.e., $\dim_{\mathbb{F}_q} W=\dim_{\mathbb{F}_q}U=$ $rn-n $,
and we give a complete characterization for $r=2$.
