Range space of Similar matrices

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Do similar matrices share the same column space $Colsp$ and null space $Nullsp$?
What I think !: If $A$ and $B$ are two similar matrices then they must have same range space and null space i.e. $Colsp(A)=Colsp(B)$ and consequently by rank-nullity theorem, we have $Nullsp(A)=Nullsp(B)$ .
I am unable to conclude the answer. Please help!!.