Understanding Homogeneous spaces

Question

I am trying to understand the homogeneous spaces. In an example of Bröcker's 'Representation theorey on Lie groups' book, it says, that the special orthogornal group $SO(n)$ operates linearly on $\mathbb{R}^n$ and transitively on the sphere $\mathbb{S}^{n-1}=\{x \in \mathbb{R}^n| |x|=1\}$. What does that mean operating linearly/ transitively? How can I see that this is the case?

Answer 1

Let $R \in SO(n)$.

$SO(n)$ acting linearly on $\mathbb R^n$ means that if $u,v \in \mathbb R^n$ and $c \in \mathbb R$ then $R(u+v) = Ru + Rv \in \mathbb R^n$ and $R(cu) = c(Ru) \in \mathbb R^n$.

I'm not sure what means $SO(n)$ acting transitively on $\mathbb S^{n-1}$ but I assume that it means that if $u \in \mathbb S^{n-1}$ then also $Ru \in \mathbb S^{n-1}$ and that if $R_1,R_2 \in SO(n)$ then there exists $R_3 \in SO(n)$ such that $R_1(R_2u) = R_3u.$