The transformations can then be understood as acting on the space of bases for V . An active transformation τ τ ∈ ∈ GL ( V ) sends the basis { e i } ↦ ↦ { τ τ e i } . Meanwhile a passive transformation T ∈ ∈ GL ( n , K ) sends the basis { e i } ↦ ↦ { ∑ ∑ j ( T − − 1 ) j i e j } .