1. The problem statement, all variables and given/known data If [itex]T:V\rightarrow V[/itex] is linear, then Ker(T^2)=Ker(T) implies Im(T^2)=Im(T). 2. Relevant equations Let [itex]T:V\rightarrow V[/itex] be a linear operator such that [itex]\forall x\in V[/itex], [itex]T^2(x)=0\Rightarrow T(x)=0[/itex] (Ker(T^2)=Ker(T)). Prove that [itex]\forall x\in V, \exists u\in V\ni T(x)=T^2(u)[/itex] (Im(T^2)=Im(T)). 3. The attempt at a solution Any clue on where I should start? I'm really stuck at this problem and have been thinking about it for the past two days. The problem is I don't know how to use the assumption Ker(T)=Ker(T^2) .