> and have an endomorphism f And the "outermorphism" f̱ of your linear transformation, when limited to considering its application to an arbitrary pseudoscalar

> and have an endomorphism f And the "outermorphism" f̱ of your linear transformation, when limited to considering its application to an arbitrary pseudoscalar, returns another pseudoscalar which necessarily has the same orientation, making that a scaling operation. So what we could say in that case is that f̱(p) / p = d (some scalar, the "determinant" of f), where p is any pseudoscalar p = v1 ∧ v2 ∧ ··· ∧ vn. This turns out to be about the same as what I wrote a few comments upthread. We are just dealing with f̱( v1 ∧ v2 ∧ ··· ∧ vn ) / ( v1 ∧ v2 ∧ ··· ∧ vn ) = d = ( f(v1) ∧ f(v2) ∧ ··· ∧ f(vn) ) / ( v1 ∧ v2 ∧ ··· ∧ vn ) instead of ( v1 ∧ v2 ∧ ··· ∧ vn ) / ( e1 ∧ e2 ∧ ··· ∧ en ) = d And now we are talking about a property of a linear transformation instead of a property of a collection of n vectors. In many practical situations, an oriented quantity like v1 ∧ v2 ∧ ··· ∧ vn is more useful than a scalar ratio d though.