Math 2.0

Linear and multilinear algebra; matrix theory Definition

Rank of a linear map

Builds on Image of a linear mapDimension

Definition

Let FF be a field, let VV and WW be vector spaces over FF, and let T ⁣:V→WT \colon V \to W be a linear map. The rank of TT is rank⁡T=dim⁡im⁡T,\rank T = \dim \im T, the dimension of the image of TT.

Remarks

For the map x↦Axx \mapsto Ax of a matrix AA, the rank is the rank of the matrix. The rank satisfies rank⁡T≤dim⁡V\rank T \le \dim V and rank⁡T≤dim⁡W\rank T \le \dim W. The rank–nullity theorem relates the rank and the nullity. The rank of a composition satisfies rank⁡(T∘S)≤min⁡(rank⁡T,rank⁡S)\rank(T \circ S) \le \min(\rank T, \rank S).

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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 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 mapdefinition
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. Linear span definition
  8. Spanning set definition
  9. Basis definition
  10. Finite-dimensional vector space definition
  11. Dimension definition
  12. Rank of a linear map definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Rank_(linear_algebra), lead section
About this page
Tag0JGB7A6GTS
URIm2://0JGB7A6GTS
Kinddefinition
WritingW1 stated
MSC15A03, 15A04
DefinesRank of a linear map
UsesImage of a linear map, Dimension

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