Let $A: \mathbb{R}^n \rightarrow \mathbb{R}^n$ a linear transformation, and let $A'$ be its hilbert adjoint.
Is it true that $\det(A) = \det(A')$?
Trying to prove:
$A A' = I$ in $\mathbb{R}^n \ \Rightarrow \ \det(A) = \pm 1$.
Since $AA' = I$ we have $\det(I) = \det(AA') = \det(A)\det(A')$, and if they're equal, then I have the result.
Thanks in advance.