This paper deals with the model companion of an endomorphism acting on a vector space, possibly with extra structure. Given a theory $T$ that $\varnothing$-defines an infinite $K$-vector space ${\mathbb{V}}$ in every model, we define $T_θ:= T \cup \{\text{``$θ$ defines a $K$-endomorphism of $\mathbb{V}$''}\}$. We then consider extensions of the form $$ T_θ\cup \big\{\sum\nolimits_{k}\bigcap\nolimits_{l}\operatorname{Ker}(ρ_{j, k, l}[θ]) = \sum\nolimits_{k}\bigcap\nolimits_{l} \operatorname{Ker}(η_{j, k, l}[θ]) : j \in \mathcal{J}\big\}, $$ where all sums and intersections are finite, and all the $ρ[θ]$'s and $η[θ]$'s are polynomials over $K$ with $θ$ plugged in. Note that properties such as $θ^2 - 2\operatorname{Id} = 0$ or $\operatorname{Ker}(θ^n) = \operatorname{Ker}(θ^{n+1})$ can be expressed in such a form. We then parametrize the consistent extensions of this form by a family $\{T^C_θ: C \in \mathcal{C}\}$ and characterize the existentially closed models of each $T^C_θ$. We also present a sufficient criterion, which depends only on $T$, for when these characterizations are first-order expressible, i.e., for when a model companion of each $T^C_θ$ exists.