Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Dimension of the dual space

Builds on Dual spaceDual basis

Statement

Let FF be a field and let VV be a finite-dimensional vector space over FF. Then the dual space V∗V^* is finite-dimensional, and dim⁡V∗=dim⁡V.\dim V^* = \dim V.

Remarks

The dual basis of a basis of VV is a basis of V∗V^* with the same number of elements. This also follows from dim⁡Hom⁡F(V,W)=dim⁡V⋅dim⁡W\dim \Hom_F(V, W) = \dim V \cdot \dim W with W=FW = F. The isomorphism V→V∗V \to V^* that sends bib_i to bi∗b_i^* depends on the basis. For infinite-dimensional VV, the dimension of V∗V^* is strictly greater than the dimension of VV.

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Vector space, definition, no evidence levelVector spacedefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionLinear independence, definition, no evidence levelLinear independencedefinitionLinear form, definition, no evidence levelLinear formdefinitionLinear span, definition, no evidence levelLinear spandefinitionSpace of linear maps, definition, no evidence levelSpace of linear mapsdefinitionDual space, definition, no evidence levelDual spacedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionBasis, definition, no evidence levelBasisdefinitionDual basis, definition, no evidence levelDual basisdefinitionDimension of the dual space, statement, no evidence levelDimension of the dual…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 map definition
  4. Linear subspace definition
  5. Linear independence definition
  6. Linear form definition
  7. Linear span definition
  8. Space of linear maps definition
  9. Dual space definition
  10. Spanning set definition
  11. Basis definition
  12. Dual basis definition
  13. Dimension of the dual space statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Dual_space, Finite-dimensional case
About this page
Tag00889YD15K
URIm2://00889YD15K
Kindstatement, proposition
WritingW0 stub
MSC15A03, 15A04
StatusThis copy of the site has no build data. Run m2 status 00889YD15K.
UsesDual space, Dual basis

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