Math 2.0

Linear and multilinear algebra; matrix theory Proof

Proof of the dimension formula for a tensor product

Builds on Basis of a tensor productDimensionBasisLinear independenceFinite-dimensional vector space

Proof

  1. Let m=dim⁡Vm = \dim V and n=dim⁡Wn = \dim W. By the definition of dimension, VV has a basis e1,…,eme_1, \dots, e_m and WW has a basis f1,…,fnf_1, \dots, f_n.
  2. By the basis of a tensor product, the family (ei⊗fj)(e_i \otimes f_j) with 1≤i≤m1 \le i \le m and 1≤j≤n1 \le j \le n is a basis of V⊗WV \otimes W.
  3. This family has mnmn indices. Its vectors are pairwise distinct, because a linearly independent family has no repeated vector. A repeat x=yx = y would give the nontrivial relation 1⋅x−1⋅y=01 \cdot x - 1 \cdot y = 0.
  4. So V⊗WV \otimes W has a basis of mnmn vectors. Hence it is finite-dimensional, and dim⁡(V⊗W)=mn=dim⁡V⋅dim⁡W\dim(V \otimes W) = mn = \dim V \cdot \dim W.

Idea

The products ei⊗fje_i \otimes f_j of basis vectors form a basis of V⊗WV \otimes W. There is one such product for each pair (i,j)(i, j), which gives dim⁡V⋅dim⁡W\dim V \cdot \dim W vectors.

Reading path

To understand this page from the ground up, read these 30 pages in order. The foundations come first. The path takes the proofs Proof that the tensor products of basis vectors form a basis, Proof that a linear map is determined by its values on a basis, Proof of the universal property of the quotient space, Proof of the universal property of the tensor product from its construction.

Show the path as a graph
Basis, concept, no evidence levelBasisconceptKernel of a linear map, concept, no evidence levelKernel of a linear mapconceptLinear combination, concept, no evidence levelLinear combinationconceptLinear map, concept, no evidence levelLinear mapconceptQuotient map, concept, no evidence levelQuotient mapconceptQuotient space, concept, no evidence levelQuotient spaceconceptVector space, definition, no evidence levelVector spacedefinitionProof that a linear map is determined by its values on a basis, proof, no evidence levelProof that a linear m…proofProof of the universal property of the quotient space, proof, no evidence levelProof of the universa…proofBilinear form, definition, no evidence levelBilinear formdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionKernel of a linear map, definition, no evidence levelKernel of a linear mapdefinitionLinear span, definition, no evidence levelLinear spandefinitionTensor product of vector spaces, definition, no evidence levelTensor product of vec…definitionQuotient space, definition, no evidence levelQuotient spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionQuotient map, definition, no evidence levelQuotient mapdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionUniversal property of the quotient space, statement, no evidence levelUniversal property of…statementDimension, definition, no evidence levelDimensiondefinitionA linear map is determined by its values on a basis, statement, no evidence levelA linear map is deter…statementProof of the universal property of the tensor product from its construction, proof, no evidence levelProof of the universa…proofUniversal property of the tensor product, statement, no evidence levelUniversal property of…statementProof that the tensor products of basis vectors form a basis, proof, no evidence levelProof that the tensor…proofBasis of a tensor product, statement, no evidence levelBasis of a tensor pro…statementProof of the dimension formula for a tensor product, proof, no evidence levelProof of the dimensio…proof
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. Basis concept
  2. Kernel of a linear map concept
  3. Linear combination concept
  4. Linear map concept
  5. Quotient map concept
  6. Quotient space concept
  7. Vector space definition
  8. Proof that a linear map is determined by its values on a basis proof
  9. Proof of the universal property of the quotient space proof
  10. Bilinear form definition
  11. Linear combination definition
  12. Linear map definition
  13. Linear subspace definition
  14. Linear independence definition
  15. Kernel of a linear map definition
  16. Linear span definition
  17. Tensor product of vector spaces definition
  18. Quotient space definition
  19. Spanning set definition
  20. Quotient map definition
  21. Basis definition
  22. Finite-dimensional vector space definition
  23. Universal property of the quotient space statement
  24. Dimension definition
  25. A linear map is determined by its values on a basis statement
  26. Proof of the universal property of the tensor product from its construction proof
  27. Universal property of the tensor product statement
  28. Proof that the tensor products of basis vectors form a basis proof
  29. Basis of a tensor product statement
  30. Proof of the dimension formula for a tensor product proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod.pdf, Theorem 4.9 and Example 4.10
About this page
Tag5EW5H4QP2T
URIm2://5EW5H4QP2T
Kindproof
WritingW0 stub
MSC15A69, 15A03
ProvesDimension of a tensor product
UsesBasis of a tensor product, Dimension, Basis, Linear independence, Finite-dimensional vector space
Changed4 November 2026

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