Math 2.0

statement theorem

An injective endomorphism of a finite-dimensional space is surjective

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.

Explanations

L1 human

For a linear map from a finite-dimensional space to itself, one to one and onto are the same thing. If no information is lost, then nothing in the target can be missed, because the dimensions match. This is like a map from a finite set to itself.

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To read this node, read these 16 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 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 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.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Rank–nullity_theorem, Reformulations and generalizations

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