  A characteristic-dependent linear rank inequality is a linear inequality that
holds by ranks of subspaces of a vector space over a finite field of determined
characteristic, and does not in general hold over other characteristics. In
this paper, we produce new characteristic-dependent linear rank inequalities by
an alternative technique to the usual Dougherty's inverse function method [9].
We take up some ideas of Blasiak [4], applied to certain complementary vector
spaces, in order to produce them. Also, we present some applications to network
coding. In particular, for each finite or co-finite set of primes $P$, we show
that there exists a sequence of networks $\mathcal{N}\left(k\right)$ in which
each member is linearly solvable over a field if and only if the characteristic
of the field is in $P$, and the linear capacity, over fields whose
characteristic is not in $P$, $\rightarrow0$ as $k\rightarrow\infty$.
