Tensor product for real numbers ?

Question

What is the tensor product of two real numbers ?

$\otimes: \mathbb R \times \mathbb R\rightarrow \mathbb R \otimes \mathbb R, \, \otimes(a,b) \mapsto a \otimes b$

I think $\mathbb R \otimes \mathbb R$ is just $\mathbb R$, no ? But I still don't understand the operation $\otimes$. What is $3 \otimes 4$ for example ? The classical multiplication ?

Answer 1

It's just regular multiplication. The tensor (more precisely the outer) product of two vectors of length $n$ is a matrix of size $n\times n$.

https://en.m.wikipedia.org/wiki/Outer_product

Answer 2

We have, for any commutative ring $R$ and $R$-module $M$ a canonical isomorphism: \begin{align} R\otimes_RM&\longrightarrow M, \\ a\otimes m&\longmapsto am \end{align} and by linearity, $\;\sum_{i=1}^n a_i\otimes m_i\longmapsto \sum_{i=1}^n a_i m_i$.