Let $V$ be an inner product space with $\dim > 1$ over field $\mathbb C$ and let $u,w \in V$.
Find $T^*$ given that $Tv=\left<v,w\right>u$
I am having a problem with this question.
I don't know how to start this question, I think it's related to the definition of $T^*$, where it's the congruent of the transpose.
$Tv=\left<v,w\right>u \Leftrightarrow \overline{Tv}=\overline{\left<v,w\right>u} \Leftrightarrow \overline T \overline v=\left<w,v\right>\overline u$
I want to apply transpose, however I can't apply transpose on both sides as I only want it applied on $T$, how do I continue?
Remember that adjoint is unique linear transformación such tha $<Tv,z>=<v, T^*z>$, so if you find another transformation which verifies that, then it must be the adjoint of T. Calculate an inner product between Tv and any vector z, use algebraic manipulations to get the form of T*. So
\begin{align*} <Tv,z> &= <<v,w>u, z > \\ & = <v,w><u,z> \\ &= <v,w> \overline{<z,u>} \\ &= \overline{<z,u>}<v,w> \\ &= <v,<z,u>w > \\ &= <v, Uz> \end{align*}
where $Uz = <z,u>w$ is a linear transformation such that $<T^*v, z> = <v, Uz>$, it follows by uniqueness of $T^*$ that $U = T^*$.