Examples of studying multilinear maps via linear maps.

Question

There is an answer of the question "Why is the tensor product important when we already have direct and semidirect products?" which states that

they allow you to study certain non linear maps (bilinear maps) by transforming them first into linear ones, to which you can apply linear algebra;---Mariano Suárez-Álvarez♦

Could anyone give me some concrete (and basic as possible, without using category theory) examples to show this application?

Please don't just show that there is a linear map which is induced by the multilinear map. Please show that how can we use the linearity of the linear map to gain some information or properties of the multilinear map.

Answer 1

In the cartesian plane geometry how do you relate the calculation that spawn two vectors? the answer is the bilinear map determinant.

Similar happens with the calculation of the volume spawn by three vectors in the 3d cartesian space.

Answer 2

Consider the Proposition 2.1.7.1 from Geometry and Complexity Theory- Landsberg.

Proposition 2.1.7.1: $\underline{R}(T) \geq \text{rank}(T_A)$

$T$ is a tensor in $A \otimes B \otimes C$, where $A,B,C$ are finite dimensional complex vector spaces. $\underline{R}(T)$ is the border rank of $T$ and $\text{rank}(T_A)$ is the rank of the linear map $T_A:A^\ast \to B \otimes C$ induced by the tensor. You can find all these definitions in the text, which is available on the internet.

Now you can ask why do we bother studying border rank?, but the thing is, the border rank is crucial to understand the complexity of matrix multiplication, which is an important open problem in mathematics. You can read something about the subject here.