Linear applications simple doubt

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Discussion Overview

The discussion revolves around the properties of injective endomorphisms in the context of linear maps, particularly focusing on whether an injective endomorphism is automatically bijective. The scope includes theoretical aspects related to linear algebra and module theory.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants assert that injective endomorphisms are bijective when considering linear maps between finite-dimensional spaces, referencing the Fredholm alternative.
  • Others question whether this conclusion follows from the rank-nullity theorem.
  • One participant provides a counterexample from module theory, noting that in the case of finitely generated modules over certain rings, an injective endomorphism may not be surjective.
  • There is a mention that while the Fredholm alternative applies in some infinite-dimensional cases, rank-nullity cannot be used in those contexts.

Areas of Agreement / Disagreement

Participants generally agree that injective endomorphisms can be bijective in finite-dimensional spaces, but there is disagreement regarding the applicability of this property in other contexts, such as finitely generated modules over different rings.

Contextual Notes

The discussion highlights limitations related to the definitions of injectivity and surjectivity in different mathematical structures, as well as the conditions under which the Fredholm alternative and rank-nullity theorem apply.

TrickyDicky
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Is an injective endomorfism automatically bijective?
 
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R136a1 said:
If you are talking about linear maps between finite-dimensional spaces, then yes. This is known as the Fredholm alternative: http://en.wikipedia.org/wiki/Fredholm_alternative

Thanks for confirming that intuition and for the pointer to Fredholm alternative, never heard of it.
 
Doesn't this follow from rank-nullity theorem?
 
for finitely generated modules over other rings this can fail, e.g. if Z is the integers, the injective endomorphism Z-->Z taking n to 3n is injective but not surjective. Interestingly however, a surjective endomorphism of a finitely generated module is always injective as well.
 
WWGD said:
Doesn't this follow from rank-nullity theorem?

Yes. But the Fredholm alternative also hold in some infinite-dimensional cases. There, you can't use rank-nullity.
 

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