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October 15th, 2013, 11:33 AM  #1 
Member Joined: Mar 2013 Posts: 71 Thanks: 4  Problem involving adjoint and orthogonal complement
Let be a linear mapping. E,F are finitedimension inner product spaces. Show that: 1) 2) 3) 4) Where: is the adjoint of A. The simbol denotes the ortogonal complement. For exemple: is the set whose elements are vectors which are orthogonal to all vectors of . I managed to prove (1) this way: 1) . Similarly to the reciprocal implication. Please, help me to prove (2), (3) and (4). 
October 16th, 2013, 01:45 AM  #2 
Senior Member Joined: Feb 2013 Posts: 281 Thanks: 0  Re: Problem involving adjoint and orthogonal complement
note1: Ortogonal complement is a symmetrical. So 4) is obvious. note2: A**=A. So 3) follows from 1) if you apply 1) to B = A*. By the way the direct proof is total similar to your clear proof, no tricks needed. 
October 16th, 2013, 04:26 AM  #3 
Member Joined: Mar 2013 Posts: 71 Thanks: 4  Re: Problem involving adjoint and orthogonal complement
Good morning, Csak. After reading your post I've tried again: (3) follows from , because (4) follows from , because . And about (2)? 
October 16th, 2013, 07:23 AM  #4 
Senior Member Joined: Feb 2013 Posts: 281 Thanks: 0  Re: Problem involving adjoint and orthogonal complement
2) follows from 3). How?

October 16th, 2013, 09:09 AM  #5 
Member Joined: Mar 2013 Posts: 71 Thanks: 4  Re: Problem involving adjoint and orthogonal complement
I have no idea, but I propose the following direct proof Obs. : such that . But . Then, . On the other hand, . But . Then, . This implies . Is that correct? 
October 16th, 2013, 10:57 AM  #6 
Senior Member Joined: Feb 2013 Posts: 281 Thanks: 0  Re: Problem involving adjoint and orthogonal complement
Simply take the ortogonal complement of 3) and you get the 2).


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