Cross product as a tensor

Question

In the three dimensions, the cross product $A \times B$ is said to be a $(1,2)$ tensor. But the definition I know about tensors is that $(1,2)$ tensor is a multilinear map $T : V^* \times V^2 \to \mathbb{R}$ where $V^*$ is the dual space of $V$. So how does the cross product fit in this definition? Could anyone explain?

Answer 1

So recall that $V^*$ is the dual space of $V$, i.e. the space of linear maps from $V$ to the ground field $\mathbb{R}$.

So what do we mean by a $(1,2)$-tensor? Well, a tensor of type $(k,l)$ is defined as an element of the vector space $\otimes_k V \bigotimes \otimes_l V^*$. Now, I claim that we can identify tensors of type $(1,l)$ with $l$-multilinear maps taking values in $V$. You are perhaps already familiar with the case $l=1$: There is an identification $\mbox{Lin}(V,V) \cong V \otimes V^*$. The identification map we want to consider for general $l$ is rather similar, and I can elaborate if you want.

Clearly then, the cross product is a bilinear map from $\mathbb{R}^3 \times \mathbb{R}^3$ to $\mathbb{R}^3$ and hence falls in the category above, for $l=2$. I hope this helps!

You can also view a tensor of type $(1,2)$ as a map in the way you described it, but this may be a little confusing. Can you explain how you came to this point of view?