Let V denote an r-dimensional Fqn-vector space. For an
m-dimensional Fq-subspace U of V assume that dimq(⟨v⟩Fqn∩U)≥2 for each
non zero vector v∈U. If n≤q then we prove the existence of an
integer 1<d∣n such that the set of one-dimensional
Fqn-subspaces generated by non-zero vectors of U is the same
as the set of one-dimensional Fqn-subspaces generated by
non-zero vectors of ⟨U⟩Fqd. If we view U as a
point set of AG(r,qn), it means that U and ⟨U⟩Fqd determine the same set of directions. We prove a
stronger statement when n∣m.
In terms of linear sets it means that an Fq-linear set of
PG(r−1,qn) has maximum field of linearity Fq only if it
has a point of weight one. We also present some consequences regarding the size
of a linear set