What is the definition of R in the tensor product construction?

[amicciulla]

Hello,

So I'm trying to understand the construction of the tensor product of 2 vector spaces as stated in the http://en.wikipedia.org/wiki/Tensor_product" . Now, in the article it states that the tensor product of two vector spaces V and W is the quotient space F( VxW )/R (F( VxW ) being the free vector space over VxW). I'm slightly confused about the definition of R, which is defined as the space generated by the 3 following equivalence relations: (v+u,w) ~ (v,w)+(u,w), (v,u+w) ~ (v,u)+(v,w), and k*(v,w) ~ (k*v,w) ~ (v,k*w). Could anybody elaborate on this? How does one generate a space from equivalence relations?

-Adam

 

[mathwonk]

you subtract the things you want to be equivalent and set those differences equal to zero. then take the space those differences generate.

if you go to my website and open up the class notes for 845-3, on page 23-28 you will find a complete discussion, and a precisely correct one.