Math 2.0

Linear and multilinear algebra; matrix theory Corollary

Dimension of a tensor product

Builds on Basis of a tensor productDimension

Statement

Let FF be a field, and let VV and WW be finite-dimensional vector spaces over FF. Then dim⁡(V⊗W)=dim⁡V⋅dim⁡W,\dim (V \otimes W) = \dim V \cdot \dim W, where dim⁡\dim is the dimension over FF.

Remarks

It follows from the basis of a tensor product, which has dim⁡V⋅dim⁡W\dim V \cdot \dim W vectors. For example, Fm⊗FnF^m \otimes F^n has dimension mnmn, like the space of m×nm \times n matrices. Compare this with the direct sum: dim⁡(V⊕W)=dim⁡V+dim⁡W\dim (V \oplus W) = \dim V + \dim W.

Reading path

To understand this page from the ground up, read these 13 pages in order. The foundations come first.

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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 setdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionBasis, definition, no evidence levelBasisdefinitionBasis of a tensor product, statement, no evidence levelBasis of a tensor pro…statementDimension, definition, no evidence levelDimensiondefinitionDimension of a tensor product, statement, no evidence levelDimension of a tensor…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.
  • no evidence level
  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. Finite-dimensional vector space definition
  10. Basis definition
  11. Basis of a tensor product statement
  12. Dimension definition
  13. Dimension of a tensor product statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Tensor_product, Dimension
About this page
Tag0R84PF4JV7
URIm2://0R84PF4JV7
Kindstatement, corollary
WritingW0 stub
MSC15A69, 15A03
StatusThis copy of the site has no build data. Run m2 status 0R84PF4JV7.
UsesBasis of a tensor product, Dimension

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