A Garden of Eden theorem for linear subshifts

Let $G$ be an amenable group and let $V$ be a finite-dimensional vector space over an arbitrary field $\K$. We prove that if $X \subset V^G$ is a strongly irreducible linear subshift of finite type and $\tau \colon X \to X$ is a linear cellular automaton, then $\tau$ is surjective if and only if it is pre-injective. We also prove that if $G$ is countable and $X \subset V^G$ is a strongly irreducible linear subshift, then every injective linear cellular automaton $\tau \colon X \to X$ is surjective.