Tensor product of vector spaces | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-29

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Tensor_product


00:02:09 1 Intuitive motivation and the concrete tensor product
00:11:09 2 Baby step toward the abstract tensor product: the free vector space
00:20:55 3 Using the free vector space to "forget" about the basis
00:23:12 4 The definition of the abstract tensor product
00:32:42 5 Properties
00:36:11 5.1 Notation
00:36:20 5.2 Dimension
00:37:47 5.3 Tensor product of linear maps
00:38:15 5.4 Universal property
00:46:54 5.5 Tensor powers and braiding
00:53:10 6 Product of tensors
00:55:55 7 Relation to dual space
01:01:09 7.1 Tensor product vs. Hom
01:03:16 7.2 Adjoint representation
01:07:03 8 Tensor products of modules over a ring
01:07:32 8.1 Tensor product of modules over a non-commutative ring
01:09:34 8.2 Computing the tensor product
01:12:40 9 Tensor product of algebras
01:19:18 10 Eigenconfigurations of tensors
01:22:18 11 Other examples of tensor products
01:24:47 11.1 Tensor product of Hilbert spaces
01:31:27 11.2 Topological tensor product
01:31:37 11.3 Tensor product of graded vector spaces
01:31:59 11.4 Tensor product of representations
01:32:25 11.5 Tensor product of quadratic forms
01:32:51 11.6 Tensor product of multilinear forms
01:33:12 11.7 Tensor product of sheaves of modules
01:33:22 11.8 Tensor product of line bundles
01:35:17 11.9 Tensor product of fields
01:35:27 11.10 Tensor product of graphs
01:35:36 11.11 Monoidal categories
01:35:46 12 Quotient algebras
01:36:18 13 Tensor product in programming
01:36:48 13.1 Array programming languages
01:41:56 14 See also
01:42:06 15 Notes
01:43:08 16 References



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SUMMARY
=======
In mathematics, the tensor product V ⊗ W of two vector spaces V and W (over the same field) is itself a vector space, endowed with the operation of bilinear composition, denoted by ⊗, from ordered pairs in the Cartesian product V × W onto V ⊗ W in a way that generalizes the outer product.  The tensor product of V and W is the vector space generated by the symbols v ⊗ w, with v ∈ V and w ∈ W, in which the relations of bilinearity are imposed for the product operation ⊗, and no other relations are assumed to hold.  The tensor product space is thus the "freest" (or most general) such vector space, in the sense of having the fewest constraints.
The tensor product of (finite dimensional) vector spaces has dimension equal to the product of the dimensions of the two factors:

  
    
      
        dim
        ⁡
        (
        V
        ⊗
        W
        )
        =
        dim
        ⁡
        V
        ×
        dim
        ⁡
        W
        .
      
    
    {\displaystyle \dim(V\otimes W)=\dim V\times \dim W.}
  In particular, this distinguishes the tensor product from the direct sum vector space, whose dimension is the sum of the dimensions of the two summands:

  
    
      
        dim
        ⁡
        (
        V
        ⊕
        W
        )
        =
        dim
        ⁡
        V
        +
        dim
        ⁡
        W
        .
      
    
    {\displaystyle \dim(V\oplus W)=\dim V+\dim W.}
  More generally, the tensor product can be extended to other categories of mathematical objects in addition to vector spaces, such as to  matrices, tensors, algebras, topological vector spaces, and modules.  In each such case the tensor product is characterized by a similar universal property: it is the freest bilinear operation.  The general concept of a "tensor product" is captured by monoidal categories; that is, the class of all things that have a tensor product is a monoidal category.