  In this paper we will study the action of $\mathbb{F}_{q^n}^{2 \times 2}$ on
the graph of an $\mathbb{F}_q$-linear function of $\mathbb{F}_{q^n}$ into
itself. In particular we will see that, under certain combinatorial
assumptions, its stabilizer (together with the sum and product of matrices) is
a field. We will also see some examples for which this does not happen.
Moreover, we will establish a connection between such a stabilizer and the
right idealizer of the rank-metric code defined by the linear function and give
some structural results in the case in which the polynomials are partially
scattered.
