On subspaces defining linear sets of maximum rank

Let V denote an r-dimensional Fₐ䂞-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. , ₅ₐ W=₅ₐU= rn-n, and we give a complete characterization for r=2.