Injective endomorphism = Surjective endomorphism

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SUMMARY

An injective endomorphism in the context of finite-dimensional vector spaces is necessarily surjective, as established by the rank-nullity theorem. This theorem states that if an endomorphism has a trivial kernel, its image achieves maximal dimension, confirming surjectivity. Conversely, a surjective endomorphism is also injective under the same finite-dimensional vector space assumption. The discussion clarifies that these properties do not hold for most other algebraic structures.

PREREQUISITES
  • Understanding of vector spaces and their properties
  • Familiarity with the rank-nullity theorem
  • Knowledge of injective and surjective functions
  • Basic concepts of linear algebra
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  • Study the rank-nullity theorem in detail
  • Explore properties of finite-dimensional vector spaces
  • Research injective and surjective mappings in linear algebra
  • Examine examples of endomorphisms in various algebraic structures
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Mathematicians, students of linear algebra, and educators seeking a deeper understanding of the relationships between injective and surjective endomorphisms in finite-dimensional vector spaces.

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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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