I am reading representations to tangent vectors of quotient manifolds from chapter 3 of Optimization on matrix manifolds, specifically trying to understand how the tangent vector representations at two points, $\bar{\xi}_x$ and $\bar{\xi}_{\alpha x}$, are related.
The real projective space $\Bbb{RP}^{n-1}$ is the quotient $\Bbb{R}^n_{*}/\sim$, where $x \sim y$ if and only if there is an $\alpha \in \Bbb{R}_*$ such that $y=\alpha x$. The equivalence class of a point $x$ of $\Bbb{R}_*$ is $$ [x]=\pi^{-1}(\pi(x))=x \mathbb{R}_{*}:=\left\{x \alpha: \alpha \in \mathbb{R}_{*}\right\} $$
The vertical space at a point $x \in \Bbb R^n_*$ is $ \mathcal{V}_{x}=x \mathbb{R}:=\{x \alpha: \alpha \in \mathbb{R}\} $
A suitable choice of horizontal distribution is $ \mathcal{H}_{x}:=\left(\mathcal{V}_{x}\right)^{\perp}:=\left\{z \in \mathbb{R}^{n}: x^{T} z=0\right\} $
Let $f: \mathbb{R} \mathbb{P}^{n-1} \rightarrow \mathbb{R}$ be an arbitrary smooth function and define $\bar{f}:=f \circ \pi: \mathbb{R}_{*}^{n} \rightarrow \mathbb{R}$. Consider the function $g: x \mapsto \alpha x,$ where $\alpha$ is an arbitrary nonzero scalar.
I was able to understand the following:
1) $\operatorname{D} g(x) \left[\bar{\xi}\right] = \alpha \bar{\xi}_x $, because $g(x)$ is a linear map.
2) $\operatorname{D} \bar{f}(g(x))\left[\operatorname{D} g(x)\left[\bar{\xi}_{x}\right]\right]=\operatorname{D} \bar{f}(x)\left[\bar{\xi}_{x}\right]$ because $\bar f(g(x)) = f(\pi(g(x))) = f(\pi(x)) =\bar f(x)$.
Using the above two facts, we can write that \begin{align*} \operatorname{D} \bar f(\alpha x)[\alpha \bar{\xi}_x] &= \operatorname{D} \bar{f}(x)\left[\bar{\xi}_{x}\right] \\ & = \operatorname{D} f(\pi(x)) \left[ \xi_x \right] \\ & = \operatorname{D} f(\pi(\alpha x)) \left[ \xi_x \right] \end{align*}
The book claims that since the above equation is valid for any smooth function $f$, this implies $\operatorname{D}\pi(\alpha x)\left[ \alpha \bar{\xi}_x \right] = \xi$. This, along with the fact that $\alpha \bar{\xi}_x$ is an element of $\mathcal{H}_{\alpha x}$, implies that $\bar{\xi}_{\alpha x}=\alpha \bar{\xi}_{x}$.
Q1) How does $ \operatorname{D} \bar f(\alpha x)[\alpha \bar{\xi}_x] = \operatorname{D} f(\pi(\alpha x)) \left[ \xi_x \right] $ imply $\operatorname{D}\pi(\alpha x)\left[ \alpha \bar{\xi}_x \right] = \xi$ ?
Q2) What is the guarantee that $\alpha \bar{\xi}_x$ belong to $\mathcal{H}_{\alpha x}$?
Edit: I think I figured out the reasoning for Q2 - $\mathcal{H}_{\alpha x} = \left\{z \in \mathbb{R}^{n}: \alpha x^{T} z=0\right\}$ - therefore $\alpha \bar{\xi}_x \in \mathcal{H}_{\alpha x}$.