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

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.

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You can find some good starting points on how to format mathematics on the site here. This AMS reference is very useful. If you need to format more advanced things, there are many excellent references on LaTeX on the internet, including StackExchange's own TeX.SE site. – Zev Chonoles Feb 22 '13 at 19:53
Thanks, Ill definately look into that later. However, could you please try to help me out with this question? – dreamer Feb 22 '13 at 19:58
@ZevChonoles Do you maybe have a clue on how I could do this question? Do you know how I can find isomorphisms such as these in general? – dreamer Feb 22 '13 at 20:45
What does $\mathsf{T}$ mean? What's your definition of the tensor product? – Chris Eagle Feb 22 '13 at 20:47
@user48288: So you have no idea what a tensor product is, but you're trying to prove things about it. Can't you see that this is a fruitless endeavour? You'd better find out your definition before doing anything else. – Chris Eagle Feb 22 '13 at 20:56

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?

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For example, see lectures on Algebra II:

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

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