multilinearen

Multilinear refers to a mathematical concept that extends the idea of linearity from single-variable functions to multiple variables. In linear algebra and related fields, a multilinear function is a function that is linear in each of its arguments when the others are held constant. This property is fundamental in various areas of mathematics, including tensor algebra, multilinear mappings, and multilinear forms.

A multilinear function of *n* variables, *f*: *V₁ × V₂ × ... × Vₙ → W*, where *V₁, V₂, ..., Vₙ* and *W* are vector spaces, satisfies linearity in each argument. For example, a bilinear function (a special case with *n* = 2) satisfies:

*f*(a₁x + a₂y, z) = a₁*f*(x, z) + a₂*f*(y, z) and

*f*(x, b₁u + b₂v) = b₁*f*(x, u) + b₂*f*(x, v)

for all scalars *a₁, a₂, b₁, b₂* and vectors *x, y ∈ V₁, u, v ∈ V₂*.

Multilinearity is widely used in defining operations on tensors, such as the tensor product, and in formulating multilinear algebra. It also appears in physics, particularly in the study of tensors in continuum mechanics and general relativity, where multilinear mappings describe transformations of physical quantities like stress or strain.

Multilinear forms are a specific type of multilinear function that map *n* vectors to a scalar (e.g., the dot product is a bilinear form). These forms are essential in defining dual spaces and in the study of determinants and volume elements in higher dimensions. The concept of multilinearity is also foundational in multilinear interpolation, a method for approximating functions using linear combinations of basis functions evaluated at multiple points.