Can someone help me with the Tensor-Product? We defined a Tensor-Product $V \otimes W$ ($V,W$ Vectorspaces) as a Quotient space $K^{V\times W} / R(V,W)$, where $R(V,W)$ is generated by the vectors $(v,\lambda w) - \lambda (v,w)$
$(v,w+w')-(v,w)-(v,w')$
$(\lambda v,w) - \lambda(v,w)$
$(v+v',w)-(v,w)-(v',w)$
and $K^{V\times W} := \{f: V\times W \rightarrow\ K\}$.
How can I imagine this Vectorspace? Each vector should be of the form $(v,w) + R(V,W)$ ? And what is the universal property?
Do not try to imagine it concretely with cosets. That is essentially useless. Tensor products are hard the first (and second?) time you see them. The only way to learn how to work with a tensor product of vector spaces is through proving basic theorems about them, so you see how you can extract properties by that universal mapping property (universal for turning bilinear maps into linear maps).