I want to prove that: [itex]Ker(T*)=[Im(T)]^\bot[/itex] Everything is in finite dimensions. What I'm trying: Let v be some vector in ImT, so there is v' so that Tv'=v. Let u be some vector in KerT*, so T*u=0. So now: <u,v>=<u,Tv'>=<T*u,v'>=0 so every vector in ImT is perpendicular to every vector in KerT*. So [itex]Ker(T*)=[Im(T)]^\bot[/itex] My intuition tells me that there is something wrong here but I can' put a finger on it.