Volume preserving linear map

Question

I know a result that volume preserving linear maps have determinant 1. How do I prove it? I understand it has to do with change of variables but does “derivative of a linear map” make sense? (Considering derivative is itself a linear map)

Answer 1

Derivative of a linear map does not always make sense but the Jacobian does and it is what appears in the variable change formula.

Answer 2

Remember that the determinant of a set of vectors is motivated to be the signed volume of the parallelepiped spanned by the vectors. So in your case the determinant of a linear map is given by the volume of the image of the unit cube under that map with some sign. From this follows trivial that if your map preserves volumes then this determinant must be $\pm1$.

In the other direction: If a linear map has determinant $\pm1$ you get that it at least preserves the volumes of cuboids. From this it is not hard to see that it preserves the volumes of all sets (by using approximation by cuboids).