amicciulla
Dec18-08, 04:41 PM
Hello,
So I'm trying to understand the construction of the tensor product of 2 vector spaces as stated in the wikipedia article (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
So I'm trying to understand the construction of the tensor product of 2 vector spaces as stated in the wikipedia article (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