Tag: tensor

Definition of Tensor Product: M Tensor N

Let R be a commutative ring with 1 and let M and N be R-modules. This blog post will be about what is the R-module M\otimes_R N. The source of the material will primarily come from Abstract Algebra, 3rd Edition (by Dummit and Foote) which is a highly recommended Algebra book for undergraduates.

Motivation

The tensor product is a construction that, roughly speaking, allows us to take “products” mn of elements m\in M and n\in N.

Definition

(Following Dummit; there is another equivalent definition using “universal property”, see Wikipedia)

M\otimes_R N is the quotient of the free \mathbb{Z}-module over M\times N (also called module of the formal linear combinations of elements of M\times N) by the subgroup generated by elements of the form:

(m_1+m_2,n)-(m_1,n)-(m_2,n)

(m,n_1+n_2)-(m,n_1)-(m,n_2)

(mr,n)-(m,rn)

The outcome of the above definition is that the following nice properties hold:

(m_1+m_2)\otimes n=m_1\otimes n+m_2\otimes n

m\otimes (n_1+n_2)=m\otimes n_1+m\otimes n_2

mr\otimes n=m\otimes rn

 

What is a Tensor?

Most people don’t encounter Tensors (the higher level advanced version of Matrices) until they reach senior undergraduate, or even graduate level.

What is a Tensor?

The best explanation I have ever seen, comes from this video by the author of A Student’s Guide to Vectors and Tensors, Daniel A. Fleisch. Using children’s blocks and laymen language, he explains what is a tensor clearly and succinctly in a way that is unbelievably crystal clear.

This YouTube video is watched over 200,000 times, a very commendable achievement for a math video!

Official Definition by Wikipedia

Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples of such relations include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of numerical values. The order (also degree) of a tensor is the dimensionality of the array needed to represent it, or equivalently, the number of indices needed to label a component of that array. For example, a linear map can be represented by a matrix (a 2-dimensional array) and therefore is a 2nd-order tensor. A vector can be represented as a 1-dimensional array and is a 1st-order tensor. Scalars are single numbers and are thus 0th-order tensors. The dimensionality of the array should not be confused with the dimension of the underlying vector space.

Cauchy stress tensor, a second-order tensor.

If you have some programming knowledge, you may view tensors as a type of multidimensional array. A more mathematical abstract way can be achieved by defining tensors in terms of elements of tensor products of vector spaces, which in turn are defined through a universal property.

Cool? The word “tensor” really strikes me as a word that is really sophisticated and complicated!