Finding an isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$

Question

Prove the isomorphism between $(U\otimes V)^{\mathsf T}$ and $B(U \times V,\mathbb{R})$, where $B$ is the collection of all bi-linear mappings. In order to do so, give a natural isomorphism between the two spaces.

My problem is that I can't really picture the two spaces. Thus, I also find it really hard to see how I could transform the one space into the other. Could anyone please help me out? It would be highly appreciated, I don't have a clue where to start.

Any help on finding isomorphisms in general would also be very welcome.

Answer 1

For example, see lectures on Algebra II:

http://www.math.toronto.edu/jkamnitz/courses/mat247/

Answer 2

It is quite hopeless to answer such a question if we do not know what is meant by the "transpose". I guess it might mean the dual vector space i.e. that you are supposed to show that

$(U\otimes{V})^{*}=Hom(U\otimes{V},\mathbb{R})\cong{B(U\times{V},\mathbb{R}})$

This is true almost by the definition of the tensor product, more specifically by its universal property. Are you familiar with that?

This universal property is as follows: for every bilinear map $\phi:U\times{V}\rightarrow{W}$ there exists a unique linear map $\psi:U\otimes{V}\rightarrow{W}$ such that $\psi\circ{f}=\phi$, where $f:U\times{V}\rightarrow{U\otimes{V}}$ is the map $(x,y)\mapsto{x\otimes{y}}$ i.e. such that $\psi(u\otimes{v})=\phi(u,v)$. That is, bilinear maps from $U\times{V}$ correspond to linear maps from $U\otimes{V}$

However, once you know this there isn't really much to prove.. Is this an exercise found in a book?