Given commutative rings R and S with a ring homomorphism f : R → S, the respective groups of invertible n × n matrices GLn(R) and GLn(S) inherit a homomorphism which we denote by GLn(f), obtained by applying f to each matrix entry.
If K is a field, then for every vector space V over K we have a "natural" injective linear map from the vector space into its double dual.