Tensor product and Brauer Group For a project I'm doing, I've been briefed to describe the Brauer Group. I know it has something to do with tensor products of algebras, but my problem is with understanding the tensor product. The textbook I was instructed to work from doesn't even explain what a tensor product of vector spaces is, it's just assumed knowledge. I've been through the first 3 pages of google (literally every link) trying to understand what the tensor product of vector spaces is, but I just can't understand it. I'm looking for a resource that explains tensor products to someone with a background in just linear algebra. If anyone could refer me to a decent resource (or give an explanation), it would be well appreciated! Thanks!
 A: You can try any standard reference in algebra to start studying tensor products. I personally recommend Dummit & Foote's Abstract Algebra ; some people recommend Lang, you can try other books. Just use a standard algebra reference, it is probably going to be in there. Tensor products are used all over the place, so any general textbook will take the time to explain it to a certain extent for sure.
If you take tensor products of finite-dimensional vector spaces over their ground field (i.e. if $V,W$ are vector spaces over the field $k$ and you consider $V \otimes_k W$), then the tensor product is very simply described ; if $V = \langle e_1,\cdots,e_n \rangle_k$ and $W = \langle f_1,\cdots,f_m \rangle_k$ (where the generating sets above are bases for the respective vector spaces), then $V \otimes W$ is the vector space with basis $\{ e_i \otimes f_j \}$. You are also given an extra multiplication map $\otimes$ which allows you to multiply elements of $V$ with elements of $W$ to obtain elements of $V \otimes W$ in a $k$-bilinear way, i.e. $k$-linear in the $V$-argument and in the $W$-argument. The choice of the bases is not relevant ; given different bases, the resulting vector space and $\otimes$-multiplication gives a canonical vector space isomorphism between the two constructions with different bases choice.
Hope that helps,
