Let V be a vector space of finite dimension over a field F. Then the following c

Question

Let V be a vector space of finite dimension over a field F. Then the following conditions on an endomorphism T of V are equivalent: T is an automorphism of V T is monic T is epic

Explanation / Answer

An endomorphism is morphism of a group to itself T: V -> V is an endomorphism 1) Automorphism is defined as an isomorphism of an object to itself Since, isomorphism are subset of morphisms. Therefore Automorphisms are subset of endomorphism. Since T is endomorphism => T is automorphism 2) Since every isomorphism is monic, and automorphism is a type of isomorphism Therefore all automorphism are monic From 1, T is endomorphism => T is automorphism => T is monic, as all automorphisms are monic 3) Since every isomorphism is epic, This is same as part 2 proof automorphism is a type of isomorphism Therefore all automorphism are monic Therefore T is endomorphism => T is automorphism => T is epic, as all automorphisms are epic

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