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I have a linear operator $\Phi$ in $\mathbb{R}^n$ and have created another operator that I believe to be its adjoint (transpose) $\Phi^T$.

What is the most direct way to verify that my $\Phi^T$ is indeed correct without having access to the matrix representation of $\Phi$ or $\Phi^T$?

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Compute $(\Phi x, y)$ and $(x, \Phi^T y)$, they should be the same.

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Oops, I mean not classical, just regular adjoint---just the direct transpose. – Steve Dec 7 '12 at 0:24
I'm not sure what you mean. You're dealing with a real vector space so adjoint and transpose are the same. – Travis Dec 7 '12 at 18:56

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