> 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 gra

> 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.