# Linear Algebra - orthogonal vector fields

1. Sep 30, 2012

### 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.

2. Oct 1, 2012

### estro

Still relevant. [Bringing back to 1st page, after 24 hours...]