  When $G$ is an arbitrary group and $V$ is a finite-dimensional vector space,
it is known that every bijective linear cellular automaton $\tau \colon V^G \to
V^G$ is reversible and that the image of every linear cellular automaton $\tau
\colon V^G \to V^G$ is closed in $V^G$ for the prodiscrete topology. In this
paper, we present a new proof of these two results which is based on the
Mittag-Leffler lemma for projective sequences of sets. We also show that if $G$
is a non-periodic group and $V$ is an infinite-dimensional vector space, then
there exist a linear cellular automaton $\tau_1 \colon V^G \to V^G$ which is
bijective but not reversible and a linear cellular automaton $\tau_2 \colon V^G
\to V^G$ whose image is not closed in $V^G$ for the prodiscrete topology.
