An active transformation is then an endomorphism on V , that is, a linear map from V to itself. Taking such a transformation τ τ ∈ ∈ End ( V ) , a vector v ∈ ∈ V transforms as v ↦ ↦ τ τ v . The components of τ τ with respect to the basis B are defined via the equation τ τ e i = ∑ ∑ j τ τ j i e j . Then, the components of v transform as v i ↦ ↦ τ τ i j v j .