Find the adjoint of an operator Let $V$ be an inner product space over $\mathbb{R}$, and $0\neq v \in V$ some vector.
$T: V \rightarrow V$ is the operator defined as: $$ T(x) = x- \frac{2\left \langle  v,x\right \rangle
}{\left \langle v,v \right \rangle
}v $$
Find the adjoint of $T$, $T^{*}$.
I tried using the definition of the adjoint, by taking some vectors $w,q$, place in $\left \langle Tw,q  \right \rangle$, and getting to a form $\left \langle  
w,T^{*}q \right \rangle$, where $T^{*}$ is written in explicit form, but i got stuck along the way.
Would appreciate any help.

We have
$$\langle Tx,y\rangle = \left\langle x - \frac{2\langle v,x\rangle}{\langle v,v\rangle}v, y\right\rangle = \langle x,y\rangle - 2\frac{\langle v,x\rangle\langle v,y\rangle}{\langle v,v\rangle} = \left\langle x,y - \frac{2\langle v,y\rangle}{\langle v,v\rangle}v\right\rangle = \langle x,Ty\rangle$$
therefore $T^* = T$.
