ABSTRACT
Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinateindependent approach, we roughly follow Emil Artin’s rigorization of determinants of matrices with entries in a field. Standard properties are derived, in particular uniqueness, from simple assumptions. We also prove existence. Soon, however, we will want to develop corresponding intrinsic versions of ideas about endomorphisms. This is multilinear algebra. Further, for example to treat the Cayley-Hamilton theorem in a forthright manner, we will want to consider modules over commutative rings, not merely vector spaces over fields.
