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## Homework Statement

Show that strictly upper triangular ##n\times n## matrices are nilpotent.

## Homework Equations

## The Attempt at a Solution

Let ##f## be the endomorphism represented by the strict upper triangular matrix ##M## in basis ##{\cal B} = (e_1,...,e_n)##.

We have that ##f(e_k) \in \text{span}(e_1,...,e_{k-1})##, ##(f\circ f)(e_k)\in f(\text{span}(e_1,...,e_{k-1}))= \text{span}(f(e_1),...,f(e_{k-1})) \subset \text{span}(e_1,...,e_{k-2}) ##... Repeating this process, we are sure that ##f^{(k)}(e_k) = 0##. So ##\ell\ge n \Rightarrow M^\ell = 0 ##, right?