# The tensor product and its motivation

• Terilien
In summary, the tensor product is a universal bilinear operation that serves as the "best" notion of multiplication for vector spaces or modules. It can be used to define other notions of a product of vectors, and it allows for a way to make bilinear maps linear. This makes it easier to handle and is a topic that is often discussed. The tensor product is expressed as the combination of two tensors, and its meaning comes from its action on two 1-forms.
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 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.

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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.

Terilien said:
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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## 1. What is the definition of the tensor product?

The tensor product is a mathematical operation that combines two vector spaces to create a new, larger vector space. It is often used in multilinear algebra and has applications in fields such as physics and engineering.

## 2. What is the motivation behind the development of the tensor product?

The tensor product was developed to provide a way to combine vector spaces in a way that preserves their essential properties, such as dimensionality, linearity, and basis independence. This allows for a more flexible and powerful approach to mathematical and scientific problems.

## 3. What are some real-world applications of the tensor product?

The tensor product has numerous applications in physics, including in the study of electromagnetism, quantum mechanics, and general relativity. It is also used in engineering for tasks such as image and signal processing, as well as in computer science for tasks such as natural language processing and machine learning.

## 4. How is the tensor product different from other vector space operations?

The tensor product differs from other vector space operations, such as addition and multiplication, in that it combines vector spaces rather than individual vectors. It also has different properties and rules, such as the ability to distribute over addition and the use of multi-index notation.

## 5. Can the tensor product be extended to more than two vector spaces?

Yes, the tensor product can be extended to any number of vector spaces. This is known as the n-fold tensor product and is denoted by the symbol ⊗n. It is defined as the result of repeatedly taking the tensor product of two vector spaces n times.

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