# Tensor-Product definition

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?

• There are numerous answers on MSE about tensor products. A quick search will probably help. However to help you to go in the right direction, suppose $V$ has basis $v_1,...,v_n$ and $W$ has basis $w_1,...,w_m$. Then a basis for $V\otimes W$ is given by the $mn$ vectors $v_i \otimes w_j$. Quotienting out by $R(V,W)$ ensures you have 'bilinearity' of sorts i.e. so that $(v_1 + av_2) \otimes w = v_1 \otimes w + a v_2 \otimes w$ and similarly in the other factor. This gives us a universal property i.e. any time we have a bilinear map from $V \times W$, we have a unique linear map from... – Osama Ghani Jul 11 '18 at 11:08
• ...$V \otimes W$ so that the bilinear map factors through the linear one. – Osama Ghani Jul 11 '18 at 11:11