Multilinear Algebra

Multilinear algebra is an extension of linear algebra, in which the study of inner products and other bilinear mappings as well as determinants is systematically embedded. The starting point is the definition of a multilinear mapping, as known from higher derivatives of differentiable functions in several variables and the determinant as a function of column vectors. The crucial idea is then the introduction of the tensor product of modules. This is a module that transforms multilinear mappings into linear mappings, thus allowing their study using methods of linear algebra. Tensor products appear in almost all areas of mathematics and play an important role especially in differential geometry, algebraic geometry, algebraic topology, and functional analysis. They are also the starting point for the derivation of various universal structures with multiplications. Prominent examples are the exterior algebra, which is a far-reaching generalization of differential forms, the symmetric algebra, and the universal enveloping algebra of a Lie algebra.