Register to reply

The tensor product and its motivation

by Terilien
Tags: motivation, product, tensor
Share this thread:
Terilien
#1
Feb19-07, 11:12 AM
P: 140
could someone please explain to me what the tensor product is and why we invented it? most resources just state it without listing a motivation.
Phys.Org News Partner Science news on Phys.org
FIXD tells car drivers via smartphone what is wrong
Team pioneers strategy for creating new materials
Team defines new biodiversity metric
Hurkyl
#2
Feb19-07, 11:32 AM
Emeritus
Sci Advisor
PF Gold
Hurkyl's Avatar
P: 16,091
It is the "best" notion of multiplication for vector spaces or modules. Any other notion of a "product" of vectors can be defined by doing something to the tensor product of the vectors.
mathwonk
#3
Feb22-07, 08:48 PM
Sci Advisor
HW Helper
mathwonk's Avatar
P: 9,486
a product is an operation which is distributive over addition. we call these bilinear operations.

a tensor product is a universal bilinear ooperatioin such that any other biklinear operation is derived from it.

i.e. if G,H are two abelian groups, there is a bilinear map GxH-->GtensH such that f=or any other bilinear map GXH-->L, THERE IS A factorizATION OF THIS MAP via GxH-->GtensH-->L.



another point f view is that the tensor product is a way of making bilinear maps linear. i.e. the factoring map GtensH-->L above is actually linear.

linearizing things is always considered a way of making them easier to handle.

pmb_phy
#4
Feb23-07, 05:58 AM
P: 2,954
The tensor product and its motivation

Quote Quote by Terilien View Post
could someone please explain to me what the tensor product is and why we invented it? most resources just state it without listing a motivation.
It appears that this topic is comming up alot.

The tensor product is a way to combine two tensors to obtain another tensor. Suppose A and B are two vectors and A is the tensor product of the two. Then the tensor product is expressed as (Note: The symbol of the tensor product is an x surrounded by a circle. Since I don't have that symbol at my disposal I will use the symbol "@" instead.)

C = A@B

The meaning of this expression comes from the action of the tensor C on two 1-forms, "m" and "n". This is defined as

C(m,n) = A@B(m,n) = A(m)B(n)


Best wishes

Pete


Register to reply

Related Discussions
Tensor Product. Linear & Abstract Algebra 11
The Tensor Product Differential Geometry 14
Tensor product Calculus & Beyond Homework 0
Tensor product Differential Geometry 17
Tensor product Differential Geometry 1