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October 15th, 2013, 11:33 AM   #1
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Problem involving adjoint and orthogonal complement

Let be a linear mapping. E,F are finite-dimension 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).
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October 16th, 2013, 01:45 AM   #2
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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.
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October 16th, 2013, 04:26 AM   #3
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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)?
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October 16th, 2013, 07:23 AM   #4
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Re: Problem involving adjoint and orthogonal complement

2) follows from 3). How?
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October 16th, 2013, 09:09 AM   #5
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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?
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October 16th, 2013, 10:57 AM   #6
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Re: Problem involving adjoint and orthogonal complement

Simply take the ortogonal complement of 3) and you get the 2).
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