Hypersurfaces in space forms satisfying the condition $L_{k}x=Ax+b$

We study hypersurfaces either in the sphere \s{n+1} or in the hyperbolic space \h{n+1} whose position vector $x$ satisfies the condition $L_{k}x=Ax+b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface for a fixed $k=0,\mathrm{.}\mathrm{.}\mathrm{.},n-1$, $A\in \mathbb{R}(n+2)\times (n+2)$ is a constant matrix and $b\in \mathbb{R}n+2$ is a constant vector. For every $k$, we prove that when $A$ is self-adjoint and $b=0$, the only hypersurfaces satisfying that condition are hypersurfaces with zero $(k+1)$-th mean curvature and constant $k$-th mean curvature, and open pieces of standard Riemannian products of the form $\text{s}m(\sqrt{1-r^2})\times \text{s}n-m(r)\subset \text{s}n+1$, with $0<r<1$, and $\text{h}m(-\sqrt{1+r^2})\times \text{s}n-m(r)\subset \text{h}n+1$, with $r>0$. If $H_k$ is constant, we also obtain a classification result for the case where $b\mathrm{\ne }0$.