  Linearized polynomials have attracted a lot of attention because of their
applications in both geometric and algebraic areas. Let $q$ be a prime power,
$n$ be a positive integer and $\sigma$ be a generator of
$\mathrm{Gal}(\mathbb{F}_{q^n}\colon\mathbb{F}_q)$. In this paper we provide
closed formulas for the coefficients of a $\sigma$-trinomial $f$ over
$\mathbb{F}_{q^n}$ which ensure that the dimension of the kernel of $f$ equals
its $\sigma$-degree, that is \emph{linearized polynomials with maximum kernel}.
As a consequence, we present explicit examples of linearized trinomials with
maximum kernel and characterize those having $\sigma$-degree $3$ and $4$. Our
techniques rely on the tools developed in [22]. Finally, we apply these results
to investigate a class of rank metric codes introduced in [8], to construct
quasi subfield polynomials and cyclic subspace codes, obtaining new solutions
to the conjecture posed in [35].
