Linear Algebra - orthogonal vector fields

[estro]

I want to prove that: Ker(T*)=[Im(T)]^\bot
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 Ker(T*)=[Im(T)]^\bot

My intuition tells me that there is something wrong here but I can' put a finger on it.

 

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