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 $\text{K}$. We prove that if $X\subset V^G$ is a strongly irreducible linear subshift of finite type and $\tau \text{\hspace{0.17em}⁣}:X\to X$ 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 $\tau \text{\hspace{0.17em}⁣}:X\to X$ is surjective.