  We consider the Grassmann graph of $k$-dimensional subspaces of an
$n$-dimensional vector space over the $q$-element field and its subgraph
$\Gamma(n,k)_q$ formed by non-degenerate linear $[n,k]_q$ codes. We assume that
$1<k<n-1$. It is well-known that every automorphism of the Grassmann graph is
induced by a semilinear automorphism of the corresponding vector space or a
semilinear isomorphism to the dual vector space; the second possibility is
realized only if $n=2k$. Our results are the following: if $q\ge 3$ or $k\ne
2$, then every isomorphism of $\Gamma(n,k)_{q}$ to a subgraph of the Grassmann
graph can be uniquely extended to an automorphism of the Grassmann graph; in
the case when $q=k=2$, there are subgraphs of the Grassmann graph isomorphic to
$\Gamma(n,k)_{q}$ and such that isomorphisms between these subgraphs and
$\Gamma(n,k)_{q}$ cannot be extended to automorphisms of the Grassmann graph.
