I'm having trouble proving whether certain group actions are transitive or not. I'm using the following definition:
A group action $\mu:G\times S \rightarrow S$ is transitive if given a point $x_0\in S$, then $\mathcal{O}_G(x_0)=S.$
I am also aware that $\mathcal{O}_G(x_0) = S$ if and only if given any two points $x,y\in S$, there exists $g\in G$ such that $gy=x$.
If I consider the group $G=\{(a,b)\vert a\in\mathbb{R}^\ast, b\in\mathbb{R} \}$ with multiplication $$(a',b')\ast(a,b) = (a'a,a'b+b')$$ with the actions $\mu_1:G\times \mathbb{R}\rightarrow \mathbb{R}$ defined by $$\mu_1((a,b),x)=ax+b$$ and $\mu_2:G\times \mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by $$\mu_2((a,b),(x,y))=(ax+by,y),$$ is correct for me to pick points $x=0$ and $(x,y)=(1,0)$ such that $$\mu_1((a,b),0)=b\Rightarrow \mathcal{O}_G(x)=\mathbb{R}\quad \text{and} \quad \mu_2((a,b),(1,0))=(a,0) \Rightarrow \mathcal{O}_G(x)\neq\mathbb{R}^2?$$ This would indicate that $\mu_1$ is a transitive action while $\mu_2$ is intransitive.