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Injective endomorphism = Surjective endomorphism

  1. Dec 15, 2013 #1
    Is an injective endomorphism necessarily surjective? And it is also true the opposite?
     
  2. jcsd
  3. Dec 15, 2013 #2
    Endomorphism of what? Groups? Vector spaces? Fields? ...
     
  4. Dec 15, 2013 #3

    jgens

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    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.
     
  5. Dec 15, 2013 #4
    Yes, it's vector space. Thank you very much for the answer.
     
  6. Dec 15, 2013 #5
    You seem to assume the vector space is finite-dimensional :wink:
     
  7. Dec 15, 2013 #6

    jgens

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    Ah true! Lots of assumptions flying around up there!
     
  8. Dec 16, 2013 #7

    ChrisVer

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