>>13461559While technically it's not an issue to have a heterogenous matrix
(after all a matrix is just a list of lists over your domain of sets and {3, R, (-6,7)} is a valid set too), in algebra you don't see heterogenous matrices often.
That said, in your case there's effectively no matrix involved - the author just generalizes the algebraic definition of the determinant to the two different types of object of your vector space.
So while a determinant over a field would give
det([[a,b]], [c,d]]) := a*d - b*c
the author here lets
det([V,W], [c,d]) := V*d - W*c
where the first pair holds vectors - and this still makes sense, since you can multiply vectors with scalars.
>>13461301iirc pushed back against some sort of affirmative action
