Math 2.0

statement proposition

Dimension of the dual space

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.

Explanations

L1 human

A linear form on an nn-dimensional space is fixed by its nn values on a basis. These nn values can be any numbers. So the linear forms make up an nn-dimensional space too.

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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 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 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. 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

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Sources

  1. Web https://en.wikipedia.org/wiki/Dual_space, Finite-dimensional case

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