Let $A\in \Bbb R^{n\times n}$ and $B \in \Bbb R ^{n\times m}$. Show that there exists a vector $p\in \Bbb R ^m$ such that $(A,Bp)$ is controllable iff $(A,B)$ is controllable.
Here when I say $(A,B)$ is controllable i mean that the system $\dot{x}=Ax +Bu$ is controllable.
First I show that if $(A,B)$ is controllable then $(A,Bp)$ is controllable also. One characterization of $(A,B)$ being controllable is that there exists a row vector $\eta \in \Bbb R^{n}$ such that $\eta e^{At}B=0$ for all $t\in [0,T]$ where $T$ is some time greater than zero. Now using this same $\eta$ we find that
$$\eta e^{At} (Bp) = (\eta e^{At} B)p =0$$
and thus $(A,Bp)$ is also controllable for any $p \in \Bbb R^m$ actually (is this right, it doesn't seem right, for instance if $p$ is all zeros the system is obviously not controllable). So this result is probably already not right, however I cannot find the mistake.
For the reverse implication I am completely stuck. I am sorry if I made any silly mistakes here, the whole topic of controllability is still new to me. If anyone could help me out I would be very thankful!
EDIT: Reading the wording of the original question again, I think we can just take $p$ to be a column vector of all ones. Then $(A,B)$ is controllable iff $(A,Bp)$ is controllable, so we have shown the existence of the vector $p$. Is this too simple?