Let F be the set of all ordered bases. Then F is a principal homogeneous space for GL(n). Let W be a vector space and let ρ ρ be a representation of GL(n) on W (that is, a group homomorphism ρ ρ : GL ( n ) → → GL ( W ) ). Then a tensor of type ρ ρ is an equivariant map T : F → → W . Equivariance here means that