Isomorphism between $\Bbb{R}^2 \times \Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2$

Question

I hope you can help me with this:

Show that $\Bbb{R}^2 \times \Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2$ are isomorphic, and specify an isomorphism.

Thanks.

Answer 1

In the beginning I just specify what Rob Arthan meant. There are following three theorems. For simplicity considered vector spaces are over filed $\mathbb{R}.$

1.If $V$ and $W$ are finite dimensional vector spaces then $\dim(V\times W)=\dim(V)+\dim(W).$

Be awere that in category of vector spaces we can use $\times$ and $\oplus$ interchangeably.

2.If $V$ and $W$ are finite dimensional vector spaces then $\dim(V\otimes W)=\dim(V)\cdot\dim(W).$

The third one

3.If $V$ and $W$ are finite dimensional vector spaces such that $\dim(V)=\dim(W)$ then $V$ and $W$ are isomorphic.

Using these three theorems to your case you see that $\Bbb{R}^2 \times \Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2$ are isomorphic because $$2+2=2\cdot 2.$$ But to construct this isomorphism you have to refer to basis in $\Bbb{R}^2 \times\Bbb{R}^2$ and $\Bbb{R}^2 \otimes \Bbb{R}^2.$ So let $e_1=(1,0)$ and $e_2=(0,1).$ Hence $$\{(e_1,0),(e_2,0),(0,e_1),(0,e_2)\}$$ is a base in $\Bbb{R}^2 \times\Bbb{R}^2$ and $$\{e_1\otimes e_1,e_1\otimes e_2,e_2\otimes e_1,e_2\otimes e_2\}$$ is a base in $\Bbb{R}^2 \otimes \Bbb{R}^2.$

Now the required isomorphism is for example $\mathbb{R}$-linear map $\phi:\Bbb{R}^2 \times\Bbb{R}^2\rightarrow \Bbb{R}^2 \otimes \Bbb{R}^2$ such that$$\phi:(e_1,0)\mapsto e_1\otimes e_1,(e_2,0)\mapsto e_2\otimes e_1,(0,e_1)\mapsto e_1\otimes e_2,(0,e_2)\mapsto e_2\otimes e_2.$$ $\phi$ is well defined, cause to define a linear map it is sufficient to determine how it acts on a basis. I left you checking that it is in fact an isomorphism.