Math 2.0

Linear and multilinear algebra; matrix theory Linear maps Theorem

Rank–nullity theorem

Builds on Rank of a linear mapNullity

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let WW be a vector space over FF. Let T ⁣:V→WT \colon V \to W be a linear map. Then dim⁡im⁡T+dim⁡ker⁡T=dim⁡V,\dim \im T + \dim \ker T = \dim V, that is, rank⁡T+nullity⁡T=dim⁡V\rank T + \operatorname{nullity} T = \dim V, with the rank and the nullity of TT.

Remarks

For an m×nm \times n matrix AA, this says rank⁡A+dim⁡{x:Ax=0}=n\rank A + \dim \{x : Ax = 0\} = n. A proof extends a basis of ker⁡T\ker T to a basis of VV; the images of the new vectors form a basis of im⁡T\im T. The theorem also follows from the first isomorphism theorem V/ker⁡T≅im⁡TV / \ker T \cong \im T.

Reading path

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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 independencedefinitionImage of a linear map, definition, no evidence levelImage of a linear mapdefinitionKernel of a linear map, definition, no evidence levelKernel of a linear mapdefinitionLinear 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 levelDimensiondefinitionRank of a linear map, definition, no evidence levelRank of a linear mapdefinitionNullity, definition, no evidence levelNullitydefinitionRank–nullity theorem, statement, no evidence levelRank–nullity theoremstatement
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. Image of a linear map definition
  7. Kernel of a linear map definition
  8. Linear span definition
  9. Spanning set definition
  10. Basis definition
  11. Finite-dimensional vector space definition
  12. Dimension definition
  13. Rank of a linear map definition
  14. Nullity definition
  15. Rank–nullity theorem statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Rank–nullity_theorem, Stating the theorem
About this page
Tag0W4FSW1TVB
URIm2://0W4FSW1TVB
Aliasesdimension-theorem-for-linear-maps
Kindstatement, theorem
WritingW1 stated
MSC15A03, 15A04
StatusThis copy of the site has no build data. Run m2 status 0W4FSW1TVB.
UsesRank of a linear map, Nullity
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