# The tensor product and its motivation

by Terilien
Tags: motivation, product, tensor
 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.
 Emeritus Sci Advisor PF Gold P: 16,101 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.
 Sci Advisor HW Helper P: 9,428 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.
P: 2,955

## The tensor product and its motivation

 Quote by Terilien 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

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