Injective endomorphism = Surjective endomorphism

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Discussion Overview

The discussion revolves around the relationship between injective and surjective endomorphisms, particularly in the context of vector spaces. Participants explore whether an injective endomorphism must also be surjective and vice versa, while considering the implications of the rank-nullity theorem.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question the specific algebraic structure being discussed, suggesting that the nature of endomorphisms may vary across groups, vector spaces, and fields.
  • One participant asserts that if the discussion is about vector spaces, then injective endomorphisms are surjective based on the rank-nullity theorem, which states that a trivial kernel implies a maximal image dimension.
  • Another participant agrees with the vector space assumption and acknowledges the previous explanation.
  • There is a suggestion that the proof provided assumes the vector space is finite-dimensional, which introduces a potential limitation to the argument.
  • Participants express awareness of the various assumptions being made in the discussion.

Areas of Agreement / Disagreement

Participants generally agree that the discussion pertains to vector spaces, but there is no consensus on whether the injective and surjective properties hold in all cases, particularly regarding dimensionality assumptions.

Contextual Notes

There are unresolved assumptions regarding the dimensionality of the vector space, which may affect the validity of the claims made about injective and surjective endomorphisms.

Daaavde
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Is an injective endomorphism necessarily surjective? And it is also true the opposite?
 
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Endomorphism of what? Groups? Vector spaces? Fields? ...
 
Based on the OPs questions from yesterday this is probably about vector spaces. The claim is false for most other algebraic structures anyway, so if the vector space assumption is correct, then this just follows from the rank-nullity theorem. In detail, the endomorphism has trivial kernel, so its image has maximal dimension. This is enough to give you surjectivity. The statement that a surjective endomorphism is necessarily injective also follows with a similar proof.
 
Yes, it's vector space. Thank you very much for the answer.
 
jgens said:
Based on the OPs questions from yesterday this is probably about vector spaces. The claim is false for most other algebraic structures anyway, so if the vector space assumption is correct, then this just follows from the rank-nullity theorem. In detail, the endomorphism has trivial kernel, so its image has maximal dimension. This is enough to give you surjectivity. The statement that a surjective endomorphism is necessarily injective also follows with a similar proof.

You seem to assume the vector space is finite-dimensional :wink:
 
R136a1 said:
You seem to assume the vector space is finite-dimensional :wink:

Ah true! Lots of assumptions flying around up there!
 

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