  Let $\mathrm{SLAut}(\mathbb{F}_{q}^{n})$ denote the group of all semilinear
isometries on $\mathbb{F}_{q}^{n}$, where $q=p^{e}$ is a prime power. In this
paper, we investigate general properties of linear codes associated with
$\sigma$ duals for $\sigma\in\mathrm{SLAut}(\mathbb{F}_{q}^{n})$. We show that
the dimension of the intersection of two linear codes can be determined by
generator matrices of such codes and their $\sigma$ duals. We also show that
the dimension of $\sigma$ hull of a linear code can be determined by a
generator matrix of it or its $\sigma$ dual. We give a characterization on
$\sigma$ dual and $\sigma$ hull of a matrix-product code. We also investigate
the intersection of a pair of matrix-product codes. We provide a necessary and
sufficient condition under which any codeword of a generalized Reed-Solomon
(GRS) code or an extended GRS code is contained in its $\sigma$ dual. As an
application, we construct eleven families of $q$-ary MDS codes with new
$\ell$-Galois hulls satisfying $2(e-\ell)\mid e$, which are not covered by the
latest papers by Cao (IEEE Trans. Inf. Theory 67(12), 7964-7984, 2021) and by
Fang et al. (Cryptogr. Commun. 14(1), 145-159, 2022) when $\ell\neq
\frac{e}{2}$.
