In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function.

Dec 3, 2025 · A basis, form, function, etc., in two or more variables is said to be multilinear if it is linear in each variable separately.

In this section we will discuss a couple of items which are frequently, but not always, covered in linear algebra courses, but which we’ll need for our treatment of multilinear algebra in §§1.1.3 – 1.1.8.

A multilinear function (also called a multilinear form) is linear in each variable separately (i.e. linear in all its arguments). In other words, it is a linear function of each variable k when all the other variables …

R ALGEBRA In this chapter we study multilinear algebra, functions of several variables that are linear in each variable s. parately. Multilinear algebra is a generalization of linear algebra since a linear …

Aug 7, 2024 · 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.

A special class of multilinear functions are alternating functions. If f is a multilinear function, W1 = : : : = Wk and f(w1; : : : ; wk) = 0 whenever wi = wj for some i 6= j, then we say that f is alternating.

A basic role in multilinear algebra is played by the concepts of a tensor product, a tensor on a vector space and a multilinear form. The applications of multilinear algebra to geometry and analysis are …

Multilinear algebra appears in the study of the mechanical response of materials to stress and strain, involving various moduli of elasticity. The term "tensor" describes elements within the multilinear …

The basic idea is that a tensor is a scalar-valued multilinear function with variables in both V and V*. Note also that by Theorem 9.4, the space of linear functions on V* is V** which we view as simply V.