Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Dimension of a direct product

Builds on Product of vector spacesDimension

Statement

Let FF be a field and let VV and WW be finite-dimensional vector spaces over FF. Then the product V×WV \times W is finite-dimensional, and dim⁡(V×W)=dim⁡V+dim⁡W,\dim(V \times W) = \dim V + \dim W, where dim⁡\dim is the dimension.

Remarks

Let (vi)(v_i) be a basis of VV and let (wj)(w_j) be a basis of WW. Then the vectors (vi,0)(v_i, 0) and (0,wj)(0, w_j) together form a basis of V×WV \times W. For example, dim⁡(ℝ2×ℝ3)=5\dim(\R^2 \times \R^3) = 5. By induction, dim⁡(V1×⋯×Vk)=dim⁡V1+⋯+dim⁡Vk\dim(V_1 \times \cdots \times V_k) = \dim V_1 + \cdots + \dim V_k.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionProduct of vector spaces, definition, no evidence levelProduct of vector spa…definitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear span, definition, no evidence levelLinear spandefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionDimension of a direct product, statement, no evidence levelDimension of a direct…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. Linear combination definition
  3. Linear subspace definition
  4. Product of vector spaces definition
  5. Linear independence definition
  6. Linear span definition
  7. Spanning set definition
  8. Basis definition
  9. Finite-dimensional vector space definition
  10. Dimension definition
  11. Dimension of a direct product statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Direct_sum_of_modules, Properties
About this page
Tag06HN47VPTH
URIm2://06HN47VPTH
Kindstatement, proposition
WritingW0 stub
MSC15A03
StatusThis copy of the site has no build data. Run m2 status 06HN47VPTH.
UsesProduct of vector spaces, Dimension

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