determinant as top exterior power

This is the coordinate-free explanation of what the determinant really is: the single number by which a linear map stretches top-dimensional oriented volume. The top exterior power Λ^n(V) of an n-dimensional space is one-dimensional — there is essentially one 'volume element' — so any linear map can only scale it, and that scale factor is the determinant.

Let V be a vector space of dimension n and T: V -> V linear. Functoriality gives an induced map Λ^n(T): Λ^n(V) -> Λ^n(V) on the (1-dimensional) top exterior power, defined on a generator by Λ^n(T)(v_1 ∧ ... ∧ v_n) = T(v_1) ∧ ... ∧ T(v_n). Since Λ^n(V) is one-dimensional, Λ^n(T) is multiplication by a unique scalar, and that scalar is by definition (and in agreement with the usual formula) det(T).

This viewpoint makes the determinant's key properties immediate and basis-free. Multiplicativity det(ST) = det(S) det(T) is just functoriality of Λ^n applied to composition; det(T) = 0 exactly when T(v_1), ..., T(v_n) are dependent, i.e. T is not invertible; and the construction needs no choice of basis, explaining why the determinant is independent of basis. The general k-th exterior power Λ^k(T) likewise produces the elementary symmetric functions of the eigenvalues.