What is a linear isomorphism between $\underset{n\times m}{\times} \mathbb{R}$ and $\mathbb{R}^n\otimes\mathbb{R}^m$?

Question

What is a linear isomorphism between $\underset{n\times m}{\times} \mathbb{R}$ and $\mathbb{R}^n\otimes\mathbb{R}^m$?

Where $n\times m :=\{(i,j):0\le n-1,0\le j \le m-1\}$.

Since $\underset{n\times m}{\times} \mathbb{R}$ consists of $nm$ copies of $\mathbb{R}$ it has dimension $nm$. Also $\mathbb{R}^n\otimes\mathbb{R}^m$ has dimension $nm$. Hence they must be isomorphic as vector spaces.

Now what is a map $\psi:\underset{n\times m}{\times} \mathbb{R}\rightarrow \mathbb{R}^n\otimes\mathbb{R}^m$ that gives this isomorphism?

So what is $\psi(w_{i,j})$? Where $w_{i,j}:n\times m \rightarrow \mathbb{R}$.

Answer 1

Let $\{e_1^n, \dots, e_n^n\}$ be the standard basis on $\mathbb{R}^n$ and $\{e_1^m, \dots, e_m^m\}$ be the standard basis on $\mathbb{R}^m$.

Then a basis of $\mathbb{R}^n \otimes \mathbb{R}^m$ is given by $\{e_i^n \otimes e_j^m: 0 \le i \le n-1, 0 \le j \le m -1 \}$.

We define $\psi(w_{i,j}) = \sum_{i,j}w_{i,j} \; e_i^n \otimes e_j^m$.