Determinants and Multilinear Forms

Two routes to the same object. The computational route (Friedberg–Insel– Spence): determinants of order 2, then order $n$ by cofactor expansion, their properties (multiplicativity, effect of row operations), and a summary characterization. The structural route (Axler): first build general bilinear/quadratic forms and general alternating multilinear forms, then show the determinant is the alternating $n$-linear form on $\mathbf{F}^n$ normalized at the identity — which is also the natural setting to introduce tensor products of vector spaces.

Outline (TODO — flesh out each)

• Determinants of order 2, and of order $n$ (cofactor expansion)
• Properties of determinants; summary of important facts
• A characterization of the determinant
• Bilinear forms and symmetric bilinear forms; quadratic forms
• Alternating multilinear forms and permutations
• Defining and constructing the determinant as an alternating form
• Tensor products — of two vector spaces, of inner product spaces, of multiple vector spaces