Confusions about the proof of representations of isometries as products of reflections.

I am reading a proof of reflection representations of isometries that fix the origin in $mathbb{R}^n$. It is a simple induction on the dimension $n$. For $f(0)=0$ we have some $vneq w=f(v)$. The reflection in the hyperplane orthogonal to $u=v-w$, $r_u$, suffices $r_uf(u)=-u$ and therefore $r_uf(v)=v$. How do we know $mathbb{R}v$ is left pointwise invariant under $r_uf$? Do we need to know that $r_uf$ is affine? *Edit: we could know that any isometry is affine without knowing whether it is a product of reflections.

Now $r_uf$ is the identity on $mathbb{R}v$. Then it seems that I need to prove $r_uf$ is also an isometry on the hyperplane that goes through $O$ and orthogonal to $mathbb{R}v$. I find this very difficult to understand. It seems to me that this is a result of the fact that $r_uf$ is the identity on $mathbb{R}v$. But I could approach this conclusion. Why is $r_uf$ an isometry on the hyperplane?

These questions perplexed me when I am reading Stillwell’s Naive Lie Theory. I tried to do some online studying but I couldn’t understand the most essential parts of the proofs:

• WimC’s answer in this thread: Proving that every isometry of $mathbb{R}^n$ is of the form of a composition of at most $n+1$ reflections
• Hagen von Eitzen’s claim that $r_uf$ is an identity on $mathbb{R}v$: reflection representation of isometry

My linear algebra background is poor, so can somebody explain this to me with some easy concepts and notations? Thanks in advance. Any help will be appreciated.