On subspaces defining linear sets of maximum rank

Let V denote an r-dimensional Fqⁿ-vector space. Let U and W be Fq-subspaces of V, LU and LW the projective points of PG\,(V,qⁿ) defined by U and W respectively. We address the problem when LW=LU under the hypothesis that U and W have maximum dimension, i.e., Fq W=FqU= $rn-n $, and we give a complete characterization for $r=2$.