Math 2.0

statement corollary

Dimension of a tensor product

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.

Explanations

L1 human

The dimension of a tensor product is the product of the dimensions. The dimension of a direct sum is their sum instead. So R2⊗R3\R^2 \otimes \R^3 has dimension 66.

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To read this node, read these 13 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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. 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

Sources

  1. Web https://en.wikipedia.org/wiki/Tensor_product, Dimension

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