Math 2.0

Linear and multilinear algebra; matrix theory Theorem

Basis of a tensor product

Builds on Tensor product of vector spacesBasis

Statement

Let FF be a field, and let VV and WW be vector spaces over FF. Let (ei)i∈I(e_i)_{i \in I} be a basis of VV and let (fj)j∈J(f_j)_{j \in J} be a basis of WW. Then the family (ei⊗fj)(i,j)∈I×J(e_i \otimes f_j)_{(i, j) \in I \times J} is a basis of the tensor product V⊗WV \otimes W.

Remarks

So each vector of V⊗WV \otimes W is uniquely ∑i,jcij ei⊗fj\sum_{i,j} c_{ij}\, e_i \otimes f_j with only finitely many nonzero cij∈Fc_{ij} \in F. For v=∑iaieiv = \sum_i a_i e_i and w=∑jbjfjw = \sum_j b_j f_j, v⊗w=∑i,jaibj ei⊗fjv \otimes w = \sum_{i,j} a_i b_j\, e_i \otimes f_j. Thus the coefficients of a vector v⊗wv \otimes w form a matrix of rank at most 11.

Reading path

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Vector space, definition, no evidence levelVector spacedefinitionBilinear form, definition, no evidence levelBilinear formdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionTensor product of vector spaces, definition, no evidence levelTensor product of vec…definitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionBasis of a tensor product, statement, no evidence levelBasis of a tensor pro…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 combination definition
  4. Linear subspace definition
  5. Tensor product of vector spaces definition
  6. Linear independence definition
  7. Linear span definition
  8. Spanning set definition
  9. Basis definition
  10. Basis of a tensor product statement

Where this leads

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Further reading

  1. Web https://en.wikipedia.org/wiki/Tensor_product, From bases
About this page
Tag0SH5VT8QYA
URIm2://0SH5VT8QYA
Kindstatement, theorem
WritingW0 stub
MSC15A69
StatusThis copy of the site has no build data. Run m2 status 0SH5VT8QYA.
UsesTensor product of vector spaces, Basis

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