© 2017 Australian Mathematical Publishing Association Inc. Let (Formula presented.) be a vector space and let (Formula presented.) denote the semigroup (under composition) of all linear transformations from (Formula presented.) into (Formula presented.). For a fixed subspace (Formula presented.) of (Formula presented.), let (Formula presented.) be the semigroup consisting of all linear transformations from (Formula presented.) into (Formula presented.). In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’, Bull. Aust. Math. Soc. 77(3) (2008), 441–453] proved that(Formula presented.) is the largest regular subsemigroup of (Formula presented.) and characterized Green’s relations on (Formula presented.). In this paper, we determine all the maximal regular subsemigroups of (Formula presented.) when (Formula presented.) is a finite-dimensional subspace of (Formula presented.) over a finite field. Moreover, we compute the rank and idempotent rank of (Formula presented.) when (Formula presented.) is an (Formula presented.)-dimensional subspace of an (Formula presented.)-dimensional vector space (Formula presented.) over a finite field (Formula presented.).