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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?

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up vote 1 down vote accepted


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.

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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 ? – user66906 Jun 15 '13 at 12:04 has to, hasn't it? I mean, that's the meaning of orthogonal complement... – DonAntonio Jun 15 '13 at 12:09

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