Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Universal property of the tensor product

Builds on Tensor product of vector spacesBilinear formLinear map

Statement

Let FF be a field, and let VV, WW and UU be vector spaces over FF. Let h ⁣:V×W→Uh \colon V \times W \to U be a bilinear map. Then there is exactly one linear map h~ ⁣:V⊗W→U\tilde h \colon V \otimes W \to U such that h~(v⊗w)=h(v,w)for all v∈V and w∈W,\tilde h(v \otimes w) = h(v, w) \qquad \text{for all } v \in V \text{ and } w \in W, where V⊗WV \otimes W is the tensor product.

Remarks

So the bilinear maps V×W→UV \times W \to U correspond one to one with the linear maps V⊗W→UV \otimes W \to U. This property determines V⊗WV \otimes W up to a unique isomorphism that fixes the vectors v⊗wv \otimes w. Uniqueness holds because the vectors v⊗wv \otimes w span V⊗WV \otimes W.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionBilinear form, definition, no evidence levelBilinear formdefinitionLinear map, definition, no evidence levelLinear mapdefinitionTensor product of vector spaces, definition, no evidence levelTensor product of vec…definitionUniversal property of the tensor product, statement, no evidence levelUniversal property of…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Vector space definition
  2. Bilinear form definition
  3. Linear map definition
  4. Tensor product of vector spaces definition
  5. Universal property of the tensor product statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Tensor_product, Universal property
About this page
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Kindstatement, theorem
WritingW0 stub
MSC15A69
StatusThis copy of the site has no build data. Run m2 status 0YDRV01R8F.
UsesTensor product of vector spaces, Bilinear form, Linear map

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