Given a transformation $T\in L(V)$, $T$ nilpotent where $V$ is a vector space of finite dimension, show that $\forall \epsilon >0$, there exists a basis $\mathcal B$ from $V$ such that $[T]_\mathcal B = (a_{ij})$ and $|a_{ij}|<\epsilon$
I have no clue on how to solve it, I tried using the rational form but it led nowhere.
I assume, as noted by Crostul, that $T$ is nilpotent.
The first step, which I assume you know, is to find a basis with respect to which the matrix of $T$ is made of Jordan blocks. Let us consider one of these blocks, so there are linearly independent vectors $$ v_{0}, v_{1}, \dots, v_{k} $$ such that $$ T(v_{0}) = v_{1}, T(v_{1}) = v_{2}, \dots, T(v_{k-1}) = v_{k}, T(v_{k}) = 0. $$ Now given any $a > 0$, consider the linearly independent vectors $$ v_{0}' = v_{0}, v_{1}' = a^{-1} v_{1}, v_{2}' = a^{-2} v_{2}, \dots, v_{k}' = a^{-k} v_{k}, $$ and see what is the matrix of the action of $T$ on the $v_{i}'$. Then choose $0 < a < \min(1, \epsilon)$.