Question: Let $R$ be a commutative ring and $A$ and $B$ two $R$-module. Suppose that $A$ is free of rank $n$ with basis $a\_1,\dots,a\_n$. Then there is an isomorphism $\Phi: Hom\_R(A,B) \to Hom\_R(A,R)\otimes\_R B$ defined by $\Phi(\sigma)=\sum\_{i=1}^n \psi\_i\otimes \sigma\_i$, where $\sigma\_i=\sigma(a\_i)$ and $\psi\_i$ is the map such that $\psi\_i(a\_j)=\delta\_{ij}$. Is there another way to describe $\Phi$?

Answer: 
There was an École d'été sur la conjecture de Birch et Swinnerton-Dyer in Paris in the summer of 2002. The [videos](http://bsd-2002.institut.math.jussieu.fr/videos/) of the lectures (by P. Colmez, H. Darmon, B. Edixhoven, B. Mazur, J. Nekovár, J. Oesterlé, W. Stein, E. Urban and S. Zhang)
are available.
I checked that the BSD conference was held at DPMMS, not the Isaac Newton Institute (INI). Quite a lot of video lectures are kept online at the INI website, and there is much to be learned there on many subjects. Similarly, the Mathematical Sciences Research Institute at Berkeley (MSRI) has many video lectures available on line. Probably other mathematical institutes around the world do this,
and the number doing so is likely to increase. So there are many opportunities available online to learn from high calibre mathematicians for those with a desire to do so. In some case, instructional conferences outlining the basic underlying theory are available too, prior to the research level conferences. Nevertheless, following such lectures usually would require a mathematician at an early stage to have previously undertaken much independent study and hard work.