A linear map T between two vector spaces V and W over the same field F satisfies

for all vectors v₁ and v₂ in V and all scalars c₁ and c₂ in F. The invertible (see invertible function) linear maps from V to V form the general linear group GL(V) under composition.

If V and W are finite-dimensional with given bases, then vectors in V and W are represented by column vectors of coordinates and T is represented by premultiplication by a matrix. The group GL(V) is isomorphic to GL(n,F), where n is the dimension of V.