Past exam of the algebra course of the University of Oxford / 2007 / A / 1 / 2 / Solution

For a matrix $v$ supported in its first row and any upper triangular matrix $a$, multiplication gives $av=a_{11}v$. Thus $Av=Kv$ has dimension at most one for every $v$, whereas the entire module has dimension three. Hence this module is not cyclic, over every field $K$.