Understanding Tensor Products: A Comprehensive Guide

[dimension10]

I've searched everywhere about tensor products but I just can't understand them. Can anyone please explain this concept to me?

 

[Bacle]

If you can tell us what you don't understand, we may help you better.

If you are talking about vector spaces, then the tensor product V(x)W
gives you a new vector space in which every bilinear map from VxW into
a third space Z becomes a linear map from V(X)W--->Z .

The existence of the tensor product follows from some algebraic lemmas
that guarantee that certain maps factor through; conditions on the kernel
of homomorphisms that allow a bilinear map VxW-->Z to factor through
V(X)W.

But if we don't know your background, or more specifically where you are
stuck, it is difficult to suggest something.

 

[dimension10]

If you can tell us what you don't understand, we may help you better.

If you are talking about vector spaces, then the tensor product V(x)W
gives you a new vector space in which every bilinear map from VxW into
a third space Z becomes a linear map from V(X)W--->Z .

The existence of the tensor product follows from some algebraic lemmas
that guarantee that certain maps factor through; conditions on the kernel
of homomorphisms that allow a bilinear map VxW-->Z to factor through
V(X)W.

But if we don't know your background, or more specifically where you are
stuck, it is difficult to suggest something.

I do not understand bilinear maps, linear maps and vector spaces. I do not even know what tensor products are about. Thanks.

 

[quasar987]

Tensor products are about linear and bilinear maps between vector spaces (in the simplest case!). And they are substantially more difficult to grasp than those. So I suggest you start by understanding linear and bilinear maps on vector space.

 

[dimension10]

Tensor products are about linear and bilinear maps between vector spaces (in the simplest case!). And they are substantially more difficult to grasp than those. So I suggest you start by understanding linear and bilinear maps on vector space.

Thanks. I managed to understand linear and bilinear vector spaces. This helped me understand tensor products, but I'm afraid I might be having a misconception. Is it true that a linear vector space is a function of 2 vectors such that the diagonal vector is a sum of the 2 vectors? Am I right to say that a bilinear vector space is a function of linear vector spaces and that a tensor product is a function of vector spaces.

Such that:

V(X)W=f(f(x,y)?

 

[quasar987]

You're neither right on wrong. None of what you wrote makes sense. Best to go back to the books.

 

[Bacle]

dimension10:

Read your definitions more carefully. A map can be linear or bilinear, but
not so for a vector space.

I don't know if you are thinking of tensoring linear maps, maybe, but
even then, you are kinda off.

 

[dimension10]

dimension10:

Read your definitions more carefully. A map can be linear or bilinear, but
not so for a vector space.

I don't know if you are thinking of tensoring linear maps, maybe, but
even then, you are kinda off.

Oh, so vector spaces have only one type? Linear?

 

[mathwonk]

a dot product is a bilinear map. a tensor product is a technical device which linearizes all bilinear maps.see my notes on my web page, or search my many posts here for this topic.

 

[Fredrik]

This thread will be useful once you have understood the definitions of the terms "vector space", "linear" and "bilinear". Start with the definitions.

Sometimes one of the terms "linear vector space" or "linear space" is used instead of "vector space", but they all mean the same thing.

 

[dimension10]

a dot product is a bilinear map. a tensor product is a technical device which linearizes all bilinear maps.


see my notes on my web page, or search my many posts here for this topic.

Oh! Thanks!