Injective endomorphism = Surjective endomorphism

[Daaavde]

Is an injective endomorphism necessarily surjective? And it is also true the opposite?

 

[R136a1]

Endomorphism of what? Groups? Vector spaces? Fields? ...

 

[jgens]

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.

 

[Daaavde]

Yes, it's vector space. Thank you very much for the answer.

 

[R136a1]

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

 

[jgens]

You seem to assume the vector space is finite-dimensional

Ah true! Lots of assumptions flying around up there!

 

[ChrisVer]

http://www.math.us.edu.pl/zkldy/zatl/szymiczek/referaty/NAKsurj.pdf