$A$ 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?

If $A^2 = 0$, then $A$ satisfies the polynomial $x^2$, and and similarly for $B$. So then the minimal polynomial for $A$ and $B$ both divide $x^2$. But since $A\neq 0$, $B \neq 0$, then $x^2$ must be the minimal polynomial for both $A$ and $B$. 
Then the Jordan Canonical Form of $A$ and $B$ will include at least one $2\times 2$ Jordan block of the form: $\left[ \begin{array}{cc}0 & 1 \\ 0 & 0 \end{array}\right]$, and if $A$ and $B$ have the same number of such blocks, they will be similar, which notably implies they would have the same rank. But you assured the dimension of the range, i.e. the rank, was the same, so they must be similar. 
