Let V be a finitelygenerated vector space over a field F and

Let V be a finitely-generated vector space over a field F and let alpha Epsilon End(V). Show that alpha is not monic if and only if there exists an endomorphism beta notequalto sigma_0 of V satisfying alpha beta = sigma_0.

Solution

Let VV be a finitely generated vector space over a field KK. Then VV has a basis.

I have a question about the proof we had in lecture.

Proof: VV is finitely generated, this means for nNnN it is span({v1,..,vn})=Vspan({v1,..,vn})=V. Now we proof the claim by induction.
n=0:n=0: The claim holds for n-1. We prove now, that the claim holds for nn: Let span({v1,..,vn})=Vspan({v1,..,vn})=V and B={v1,..,vn}B={v1,..,vn}. If BB is linear independend, BB is a basis of VV. If BBis not linear independend, there is a i0{1,..,n}i0{1,..,n} such that vi0=k=1,ki0nii0vivi0=k=1,ki0nii0vi. Therefore it it vi0{v1,..,vn}{vi0}vi0{v1,..,vn}{vi0} wich is a contradiction. So BB must be lineary independend and so BB is a basis of VV. Q.E.D.

My Question is about the end of the proof starting from \"Therefore it it vi0{v1,..,vn}{vi0}