$A$ and $B$ similar if $A^2=B^2=0$ and dimension of range $A$ and $B$ are equal Suppose $A$ and $B$ are linear transformations on finite dimensional vector space $V$,s.t. $A,B\neq 0$ and $A^2=B^2=0$. Suppose the dimension of range $A$ and $B$ are equal, can $A$ and $B$ be similar?

that is surely possible. Take for example $A= \begin{pmatrix}0 & 0 \\ 1& 0 \end{pmatrix}$ and $ B=\begin{pmatrix}0 & 1 \\ 0& 0 \end{pmatrix}$ and then it holds that
$$
A^2=B^2=0
$$
and
$$
A=SBS^{-1} \text{ with } S=\begin{pmatrix}0 & 1 \\ 1& 0 \end{pmatrix}
$$
so $A$ and $B$ are indeed similar. 
