Say I have a sequence of linear operators $A_1,...,A_n$ on a (real) vector space $V_1$. I suspect that there's a second vector space $V_2$, and an operator $A$ on $V_1\oplus V_2$, such that $A_i=A^i|_{V_1}$ for $i=1,...,n$.

Is there a way to confirm my suspicion, with or without finding such $A$ and $V_2$?

Say we confirmed my suspicion, and an oracle gave us $V_2$ along with an operator $\Delta$, and they told us that $A=C\Delta$ for some operator $C$ that acts diagonally on $V_1\oplus V_2$. They even gave us bases for $V_1,V_2$, on which $C$ also acts diagonally. How would you find $C$?