Math 2.0

Linear and multilinear algebra; matrix theory Proposition

Dimension of the annihilator

Builds on Annihilator of a subspaceDimension

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let WW be a subspace of VV. Then dim⁡W+dim⁡W0=dim⁡V,\dim W + \dim W^0 = \dim V, where W0⊆V∗W^0 \subseteq V^* is the annihilator of WW.

Remarks

A proof extends a basis b1,…,bkb_1, \dots, b_k of WW to a basis b1,…,bnb_1, \dots, b_n of VV. Then the dual basis forms bk+1∗,…,bn∗b_{k+1}^*, \dots, b_n^* are a basis of W0W^0. Another proof uses the isomorphism W0≅(V/W)∗W^0 \cong (V / W)^*. With the double dual isomorphism, it gives (W0)0=W(W^0)^0 = W in finite dimensions.

Reading path

To understand this page from the ground up, read these 15 pages in order. The foundations come first.

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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 levelBasisdefinitionAnnihilator of a subspace, definition, no evidence levelAnnihilator of a subs…definitionFinite-dimensional vector space, definition, no evidence levelFinite-dimensional ve…definitionDimension, definition, no evidence levelDimensiondefinitionDimension of the annihilator, statement, no evidence levelDimension of the anni…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. Annihilator of a subspace definition
  13. Finite-dimensional vector space definition
  14. Dimension definition
  15. Dimension of the annihilator statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Dual_space, Quotient spaces and annihilators
About this page
Tag06Q1EFVH0Q
URIm2://06Q1EFVH0Q
Kindstatement, proposition
WritingW0 stub
MSC15A03, 15A04
StatusThis copy of the site has no build data. Run m2 status 06Q1EFVH0Q.
UsesAnnihilator of a subspace, Dimension

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