I'm trying to figure out exactly what the tensor product of vector spaces is. This is what I understand so far:
If $V, W$ are vector spaces over a field $R$ then the free vector space $C(V\times W)$ is a vector space which has an infinite basis (one element for each pair $(v,w)$ such that $v\in V, w\in W$. Then let the subgroup $Z$ of $C(V\times W)$ be generated by elements of the form:
1) $(v, w_1 + w_2)-(v,w_1)-(v,w_2)$
Where $a\in R$, $v \in V$, $w \in W$. The tensor product $V\otimes W$ is the quotient group $C(V\times W)/Z$.
Apparently this group now obeys the rules $(v, w_1 + w_2)-(v,w_1)-(v,w_2)=0$, and the other corresponding rules from the above, and this follows from the definition of the quotient. I haven't seen this explained anywhere and it's not immediately apparent to me at any rate. Thanks for any replies!