We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k<n$, $A$ is a constant matrix and $b$ is a constant vector.
We show that the only hypersurfaces satisfying that condition are hypersurfaces with zero $H_{k+1}$ and constant $H_k$ ( when $cneq 0$ ), open pieces of totally umbilic hypersurfaces and open pieces of the standard Riemannian product of two totally umbilic hypersurfaces.
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