this seems like a simple question but I don't understand it.
We define a transformation $T(x,y) = (y,-x)$. We want to work out what the adjoint is.
I know the answer: $T^*(x,y) = (-y,x)$ but how?
Is it because $T(x,y)$ is an operation on the first component $x$ and $T^*(x,y)$ is an operation on the second component $y$?
Thank you!
The adjoint of $T$ is defined as the operator $T^*$ such that $$ \langle T(x,y),(a,b)\rangle=\langle (x,y),T^*(a,b)\rangle $$
Let $T^*(a,b)=(u,v)$, from the definition of $T$ you have: $$ \langle (y,-x),(a,b)\rangle=\langle (x,y),(u,v)\rangle \iff ay-bx=ux+vy $$ since this have to be true for all $(x,y)$ you find: $u=-b$ and $v=a$, so: $T^*(a,b)=(-b,a)$.