controllability in two dimension

Question

Suppose my state space is $\mathbb{R}^2$, and I have a one dimensional controllable subspace $S$, my question is: is $\mathbb{R}^2/S=\{x+S:x\in\mathbb{R}^2\}$ is controllable?

Answer 1

Subspaces are not controllable or observable, Systems are.

If you have a system with state space the whole $\mathbb{R}^2$ with a controllable subspace $S_c$ i.e. $S_c=\mathcal{R}(C)=\mathcal{R}([B...A^{n-1}B])$ with $\dim(S)=r(C)=1$ and thus a uncontrollable subspace $S_{nc}, \dim(S)=2-1=1$ then the set $T=\{x+y,x \in \mathbb{R}^2, y \in S_c\}$ is again spanning $\mathbb{R}^2$, and assuming too much, this is clearly out of the controllable subspace of the original system.