Hypersurfaces in space forms satisfying the condition $L_kx=Ax+b$

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 $L_kx=Ax+b$, where $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the hypersurface for a fixed $k=0,...,n-1$, $A\in\R{(n+2)\times (n+2)}$ is a constant matrix and $b\in\R{n+2}$ 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 $\s{m}(\sqrt{1-r^2})\times\s{n-m}(r)\subset\s{n+1}$, with $0<r<1$, and $\h{m}(-\sqrt{1+r^2})\times\s{n-m}(r)\subset\h{n+1}$, with $r>0$. If $H_k$ is constant, we also obtain a classification result for the case where $b\neq 0$.