Difference between tensor product and direct product?

["pi"mp]

Hi,
I have been learning about tensor products from Dummit and Foote's Abstract Algebra and I'm a little confused. I understand the construction of going to the larger free group and "modding out" by the relations that will eventually end up giving us module structure.

But just in the case of taking the tensor product of two vector spaces to get another vector space, how is this different than just taking the direct product of the two spaces? Because the direct product will also give us a larger, new vector space.

I'm guessing the answer to my question will be along the lines of "it's because of the Universal Property the tensor product satisfies..." but I can afford to be set a little straight here!

Thanks

 

[micromass]

They are both larger spaces (for vector spaces at least), but they have different dimensions. If $V$ and $W$ are vector spaces, then the dimension of $V\times W$ is dim(V) + dim(W).
However, the dimension of $V\otimes W$ is dim(V)dim(W). So that's one difference.

The use of the tensor product is indeed in the universal property. That says that bilinear maps $V\times W\rightarrow \mathbb{R}$ correspond exactly to linear maps $V\otimes W\rightarrow \mathbb{R}$.

 

[Bacle2]

But the direct product does not come with an explicit construction as a vector space (altho it can be made into one), while the tensor product does. And the tensor is not strictly larger, at least not up to isomorphism; R(x)R~R ; it is just not of lower dimension,since the dimensions multiply.

Notice tensors are not only defined for vector spaces.