# What does the $\times$ in the context of multilinear forms in linear algebra?

I frequently see notation similar to $B: V \times V \to K$ in the context of linear algebra, where we now have the $\times$ symbol and more than one vector space. I realise that this is used to represent a linear transformation in multilinear algebra (the aforementioned notation is a bilinear form). However, my only understanding of the $\times$ symbol/operator is as the cross/Cartesian product, so I have been unable to understand what the presence of the $\times$ means in this context.

My immediate interpretation is that $B: V \times V \to K$ denotes a linear transformation $B$ that maps from a vector space which is the cross product of $V$ with itself to another vector space, $K$. However, I don't even understand what the cross product of two vector spaces means.

I would greatly appreciate it if people could please take the time to explain what this notation actually means. My studies until now have focused on what would likely be considered elementary linear algebra, so I would appreciate an explanation that is targeted towards this level of knowledge.

EDIT: I have also seen the notation $\mathbf{R}^n \times \mathbf{R}^n$ (specifically, second order tensors $\in \mathbf{R}^n \times \mathbf{R}^n$), so I would also appreciate an explanation of what this means.

• It is the Cartesian product of sets. – Lord Shark the Unknown Oct 5 '17 at 5:16

## 1 Answer

Functions always have exactly one argument. When we want to express that a function $f$ takes $k$ arguments, the common convention is therefore to let $f$ have $k$-tuples as arguments, and this is where Cartesian products are used.

$V \times V$ is the Cartesian product of $V$ with itself, that is,

$$V \times V = \{ (v_1,v_2) \mid v_1 \in V, v_2 \in V \}$$

A bilinear form $B: V \times V \rightarrow K$ (where $K$ denotes the field of scalars) takes two vectors and returns a value from $K$ – and is linear in each of its arguments.