Dimension of isotropy group in homogenous space

Question

Just sincerely ask a fundamental problem as the following:

We know

1. $G = GL(2,\mathbb{R})$ is a transitive group of dimension $4$.
2. The vector space $GL(2,\mathbb{R})$ acting on is $X=\mathbb{R}^2$, which is homogenous space.
3. Since $G$ is a transitive group, $X$ is isomorphic to $G/G_x$, where $G_x$ is an isotropy group. So $\dim G/G_x=\dim X = 2$
4. $\dim G/G_x = \dim G-\dim G_x\Rightarrow 2 = 4-\dim G_x$.
5. So I get $\dim G_x = 2$.

Am I correct? Or is there any step wrong?

Thanks