Given an infinite-dimensional vector space V, we consider the semigroup GS (m, n) consisting of all injective linear α: V → V for which codim ran α = n, where dim V = m ≥ n ≥ ℵ0. This is a linear version of the well-known Baer–Levi semigroup BL (p, q) defined on an infinite set X, where |X| = p ≥ q ≥ ℵ0. We show that, although the basic properties of GS (m, n) are the same as those of BL (p, q), the two semigroups are never isomorphic. We also determine all left ideals of GS (m, n) and some of its maximal subsemigroups; in this, we follow previous work on BL (p, q) by Sutov and Sullivan as well as Levi and Wood.