Math 2.0

Linear and multilinear algebra; matrix theory Theorem

An injective endomorphism of a finite-dimensional space is surjective

Builds on Rank–nullity theorem

Statement

Let FF be a field, let VV be a finite-dimensional vector space over FF, and let T ⁣:V→VT \colon V \to V be a linear map. Then these conditions are equivalent:

  1. TT is injective.
  2. TT is surjective.
  3. TT is bijective.

Remarks

The proof uses the rank–nullity theorem: ker⁡T=0\ker T = 0 if and only if dim⁡im⁡T=dim⁡V\dim \im T = \dim V. It fails in infinite dimensions: on F[x]F[x], multiplication by xx is injective and not surjective. In characteristic 00, the derivative on F[x]F[x] is surjective and not injective. For square matrices, this is the statement that Ax=0Ax = 0 has only the trivial solution if and only if Ax=bAx = b has a solution for each bb.

Reading path

To understand this page from the ground up, read these 16 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 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 theoremstatementAn injective endomorphism of a finite-dimensional space is surjective, statement, no evidence levelAn injective endomorp…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.
  • no evidence level
  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
  16. An injective endomorphism of a finite-dimensional space is surjective statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Rank–nullity_theorem, Reformulations and generalizations
About this page
Tag0912EEEVCT
URIm2://0912EEEVCT
Kindstatement, theorem
WritingW0 stub
MSC15A04
StatusThis copy of the site has no build data. Run m2 status 0912EEEVCT.
UsesRank–nullity theorem

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