 Multilinear algebra

Multilinear algebra studies multilinear maps, which are linear in each input separately while the other inputs stay fixed. Bilinear forms, determinants, and tensor operations are basic examples.

For instance, $B(x,y)=x^{T}Ay$ is bilinear, but $x\mapsto B(x,x)$ is a quadratic form rather than a linear map. The tensor product replaces multilinear dependence by linear dependence on a larger space: each bilinear map $V\times W\to U$ factors uniquely through a linear map $V\otimes W\to U$. Symmetric powers and exterior powers then impose symmetry or alternation, organizing polynomial and oriented-volume calculations in a coordinate-independent way.

Multilinear algebra is a branch of mathematics that extends linear algebra by dealing with multilinear functions, which are functions that are linear in each of several arguments. This area of study is essential for understanding vector spaces and can be thought of as a natural progression from linear algebra into more complex structures.