I have just begun learning about tensor products. Given are the following relations of the tensor product between vector spaces $A$ and $B$:
- $\lambda(a\otimes b)=(\lambda a)\otimes b=a\otimes(\lambda b) $ for $\lambda \in \mathbb{R}$.
- $a\otimes b_1 + a\otimes b_2 = a\otimes(b_1+b_2)$.
- $a_1\otimes b + a_2\otimes b=(a_1+a_2)\otimes b$.
For $a\in A$ and $b\in B$.
Now I want to show that $\mathbb{R}\otimes\mathbb{R}\cong \mathbb{R}$, but I am not sure where to start. Is it necessary to explicitly construct an isomorphism between the two spaces? Is there something else I should know about tensor products to get this started?