Tensor product of vector spaces

In summary, the conversation discusses the concept of tensor product of vector spaces and how to achieve it without assuming finite dimensions or a specific underlying field. The possibility of defining it without using a basis of the vector spaces is also mentioned. A helpful thread with posts 1, 2, 8, and 9 is suggested as a resource, noting a correction made in post 1.
  • #1
ShayanJ
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I'm trying to understand tensor product of vector spaces and how it is done,but looks like nothing that I read,helps!
I need to know how can we achieve the tensor product of two vector spaces without getting specific by e.g. assuming finite dimensions or any specific underlying field.
Another thing,is it possible to define it with no reference to any basis of the vector spaces?
Thanks
 
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  • #2
This thread should be useful. See in particular posts 1,2,8,9. Note that when quasar987 corrects a mistake I made in post 1, he's referring to a mistake that was later fixed in an edit. So post 1 should be OK.
 

1. What is the tensor product of vector spaces?

The tensor product of vector spaces is a mathematical operation that combines two vector spaces to form a new vector space. It is denoted by the symbol ⊗ and is used to represent the outer product of two vectors.

2. How is the tensor product represented mathematically?

The tensor product of two vector spaces V and W is represented as V ⊗ W and is defined as the set of all possible linear combinations of the elements of V and W. It is also known as the direct product of two vector spaces.

3. What is the purpose of the tensor product in linear algebra?

The tensor product is used in linear algebra to generalize the concept of the outer product and to define multilinear transformations. It is also used in various other fields of mathematics, such as differential geometry and quantum mechanics.

4. How is the tensor product calculated?

The tensor product of two vector spaces can be calculated by taking the Cartesian product of the two vector spaces and then defining a suitable bilinear operation on the resulting set. This operation must satisfy certain properties, such as associativity and distributivity, to ensure that the resulting set is also a vector space.

5. What are some applications of the tensor product in science and engineering?

The tensor product is used in various applications in science and engineering, such as image processing, signal processing, and machine learning. It is also used in physics to describe the relationship between different physical quantities, such as force and displacement.

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