Question about defining maps on Tensor Product Space Let $V,W$ be two real valued finite dimensional vector-spaces. In our lecture we defined the tensor product space $V \otimes W$ by considering the free group $\mathbb{R}^{V \times W}$ and factoring out a subset $H$ which containts elemts of the form $(v+w,z)-(v,z)-(w,z)$ etc.
Now suppose I want to define a linear map $f \colon V \otimes W \to Z$, where $Z$ is another real valued finite dimensional vector space. I could define this map on so called simple tensors $v \otimes w$, but as we know not every element is a simple tensor. How do I extend this to whole $V \otimes W$?
Is it by considering $F \colon V \times W \to Z$ which basically does the same and then using the universal property? That would tell me, that there exists a map $f$ on $V \otimes W$ such that:
$f(\phi(v,w))=F(v,w)$, where $\phi$ is the function assigned to the tensor product. How do I know that $\phi(v,w)= v \otimes w$?
Thanks in advance
 A: $v\otimes w$ is the definition of $\phi(v,w)$. The tensor product is a vector space $V\otimes_FW$ together with a bilinear map $\phi:V\times W\to V\otimes_FW$ which satisfy the following universal property: if $f:V\times W\to Z$ is any bilinear map then there is a unique linear transformation $g:V\otimes_FW\to Z$ such that $f=g\circ\phi$. We denote $\phi(v,w)=v\otimes w$.
Specifically in the construction of the tensor product which you presented, $\phi$ is the quotient map.
Now, the statement "it is easy to define a map on simple tensors" is not true. Actually, it is very not easy to say when do we have an equality of the form $v_1\otimes w_1=v_2\otimes w_2$. So most of the time it's not a good idea to define a linear map on the tensor product directly. Better to use the universal property. First define a bilinear map $f:V\times W\to Z$, and then using the universal property you get a linear map $g:V\otimes_FW\to Z$ such that $g(v\otimes w)=f(v,w)$. Every element in $V\otimes_FW$ is a finite sum of simple tensors, so you also know how $g$ is defined on the other elements.
