Find conjugate transpose (adjoint) of linear transform

Question

A difficult question I've been trying to tackle but I seem to hit a dead end.

Let $V$ be an inner product space over $\mathbb R$. We are required to find $T^{*}$ such that $$\left<T(u),v\right> = \left<u,T^{*}(v)\right>$$ where:

$$T(u) = u-\frac{2\left<v,u\right>}{\left<v,v\right>}v$$ and $v$ is some vector in $V$.

Answer 1

It's simple to see that (just calculate it) $$\langle T(u),v\rangle=\langle u-2\frac{\langle v,u\rangle}{\langle v,v\rangle}v,v\rangle=\langle u,v\rangle-2\frac{\langle v,u\rangle}{\langle v,v\rangle}\langle v,v\rangle=-\langle u,v\rangle=\langle u,-\mathrm{id}(v)\rangle$$ so $T^*=-\mathrm{id}$.