>>13110137An "abstract" determinant, more often called a volume form in this context, on an n-dimensional vector space V is a function D which takes n vectors in V as inputs and gives back a real number as an output such that:
- D is multilinear
- D is alternating (meaning D(u,u,-,-,..,-) = 0)
There are infinitely many such functions, in fact they form a one-dimensional vector space. If V is equipped with a dot product and an orientation, then there's one more condition which defines D uniquely: we require that D gives 1 on an orthonormal and positively oriented basis (in R^n this says that D of the identity matrix is 1).
Intuition is that the conditions (multilinearity and antisymmetry) capture the idea of an oriented volume of a parallelepiped spanned by the inputs, the normalization condition than says that the cube spanned an positive orthonormal basis has oriented volume 1.
