We present a detailed study on the strain-induced magnetic domain structure of a (La,Sr)MnO3 thin film epitaxially grown on a BaTiO3 substrate through the use of polarization-dependent X-ray photoemission electron microscopy and X-ray absorption spectroscopy. Angular-dependent measurements allow us to detect vector magnetization on a single-domain scale, and we relate the strain-induced changes in magnetic anisotropy of the ferromagnetic film to the ferroelectric domain structure of the underlying substrate using X-ray magnetic circular and linear dichroism spectro-microscopy. Comparisons to measurements on a nearly strain free film of (La,Sr)MnO3 grown on a (La,Sr)(Al,Ta)O3 substrate illustrate that the BaTiO3 ferroelectric domain structure imprints specific domain sizes and wall orientations in the (La,Sr)MnO3/BaTiO3 artificial multiferroic heterostructure. Furthermore, a change of the BaTiO3 ferroelectric domain structure either with temperature or with applied electric field results in a corresponding change in the (La,Sr)MnO3 ferromagnetic domain structure, thus showing a possible route to obtain room-temperature electric field control of magnetic anisotropy at the nanoscale.