Take the tour ×  I'm trying to find a way to prove that the
determinant of the transpose of an endomorphism is the
determinant of the original linear map (i.e. det(A) = det(Aᵀ) in
matrix language) using Dieudonne's definition of the determinant
expressed in terms of an alternating bilinear form but am having
problems with it. Let me set the stage a little:  Start with a
Bilinear form ψ : V × V → ℝ | (u,v) ↦ ψ(u,v) =
ψ(∑ᵢ₌₁²λᵢeᵢ,∑ᵢ₌₁²μᵢeᵢ).  It is alternating if ψ(v,v) = 0 & anti-
symmetry, ψ(u,v) = - ψ(v,u), follows by expanding ψ(u + v,u + v)
= 0  Consider the endomorphism f : V → V where V is finite-
dimensional (in this case, 2-D). If ψ(f(u),f(v)) = γψ(u,v) for
some γ ∈ ℝ (a unique γ which follows by expanding the
alternating form & showing γ is determined by the action of ψ on
it's basis) & we intuitively interpret the alternating bilinear
form ψ(u,v) as the area spanned by u & v the geometric
interpretation of γ is inescapably obvious. Here γ is known as
the determinant of the map f, i.e. ψ(f(u),f(v)) = de