Why does $ (A^T x)· y = x ·(A y) $ hold?
The proof has to do with properties of transposes. I did a proof using coordinates (which was correct) but there is an infinitely easier way to do it.
A is an n by n matrix.
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Too long for a comment:
It doesn't...and I think there is some inner product and there're some assumptions on $\,A\,$, since if we denote by $\,\langle\,x,y\,\rangle\,$ the inner product of $\,x,y\,$ , then
where $\,A^*\,$ is the adjoint of $\,A\,$ so if this is what is meant in your link and if $\, A^*=A^t\,$ then we're done (for example, if the linear space is real and we're working with an orthonormal basis...)