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If you define determinant as volume, how do you define volume? I agree that it's pedagogically sound to motivate the notion of determinant by the volume of a pa
by The_suffocated 6y ago
If you define determinant as volume, how do you define volume? I agree that it's pedagogically sound to motivate the notion of determinant by the volume of a parallelepiped, but using volume as the definition of determinant just doesn't sound right.
And strictly speaking, determinant is not volume because the former is dimensionless. It is the scaling factor of the volume when a geometric entity is transformed by a linear map.
- enriquto 6y ago> If you define determinant as volume, how do you define volume? How do you define "length" and "area"? I guess that if you don't have already a very firm grasp of these basic concepts, then there's no business for you (yet) in studying determinants. Much later, once you master thoroughly lengths, areas, volumes and hypervolumes; and also linear algebra and determinants (however they are defined), then you can embark in the elegant definitions using exterior algebra and the like. Notice that Halmos itself says that his treatment is appropriate for a *second* course in linear algebra, preparing the field for the later study of infinite-dimensional spaces. > And strictly speaking, determinant is not volume because the former is dimensionless. This really depends on the context. If you are working on euclidean space, you already have "units" and the determinant makes sense in itself, as the volume spanned by sets of vectors.