Determinant: Alternative Definitions Reference
Foundation for: Determinant: Continuity
Problem
Given a vector space $V$.
Consider an endomorphism $T:V\to V$.
The rank of an endomorphism:
$$\mathrm{rank}T:=\dim\left(\mathrm{im}T\right)$$
The determinant of an endomorphism:
$$\det T:=\text{???}$$
What would be a nice definition not relying on representations by matrices?
(I assume basic knowledge of Differential Geometry and Functional Analysis.)

There is no problem to write a definition of the determinant functional τ using induction by n and avoiding any mention of either matrices or n-forms. But it will not be very convenient since proof of uniqueness should rely on (concealed) decomposition of a matrix to upper and lower triangular factors.
This set of 3 axioms is sufficient to define unique τ on finite-dimensional spaces:
(Multiplicativity) $τ(AB) = τ(A)\,τ(B)$ for any endomorphisms A, B of V.
(Normalization) If $\dim V = 1$, then $τ(λI) = λ$.
(Decomposition) Let $U\subset V$ be a non-trivial (proper and ≠ {0}) A-invariant subspace (i.e. $\operatorname{im}(A\vert_U)\subseteq U$); we can then define the quotient space $W = V / U$ and such unique endomorphism $\hat A: W\to W$ that the diagram
 V → W
 ↓A   ↓$\hat A$
 V → W
commutes. Then $τ(A) = τ_U(A\vert_U)\,τ_W(\hat A)$ must hold, where right-hand side terms refer to τ defined on the subspace and the quotient space.
Of course, one might prove existence of τ for V of any finite dimension then, that is not a very easy task.
