Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I am struggling with the following excercise:

Let A be a matrix, then we have for every subspace $U$ that:

$A^*(U ^\perp)=(A^{-1}(U))^\perp$

I do not even know where to start to solve this excercise. Does anybody have a hint for me?

share|improve this question

1 Answer 1

up vote 1 down vote accepted

Hints:

We have

$$w\in U^\perp\;\wedge v\in A^{-1}(U)\;(\text{so}\;\;v=A^{-1}u\;\;\text{for some}\;\;u\in U\;):$$

$$\;\langle A^*w,v\rangle=\langle w,Av\rangle=\langle w,u\rangle=0$$

and thus we get $\;A^*(U^\perp)\subset \left(A^{-1}(U)\right)^\perp\;$ .

Now you try to prove the other direction inclusion.

share|improve this answer
    
does for $v=A^{-1}u$ and $p \in (A^{-1}(U))^\perp$ $0=\langle A^{-1}u,p \rangle = \langle u, (A^*(p))^{-1} \rangle $ work ? –  user180097 Jun 15 '13 at 12:04
    
Well...it has to, hasn't it? I mean, that's the meaning of orthogonal complement... –  DonAntonio Jun 15 '13 at 12:09

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.