Gravitational interactions between the Large Magellanic Cloud (LMC) and the stellar and dark matter halo of the Milky Way are expected to give rise to disequilibrium phenomena in the outer Milky Way1-7. A local wake is predicted to trail the orbit of the LMC, and a large-scale overdensity is predicted to exist across a large area of the northern Galactic hemisphere. Here we report the detection of both the local wake and northern overdensity (hereafter the `collective response') in a map of the Galaxy based on 1,301 stars at Galactocentric distances between 60 and 100 kiloparsecs. The location of the wake is in good agreement with an N-body simulation that includes the dynamical effect of the LMC on the Milky Way halo. The density contrast of the wake and collective response are stronger in the data than in the simulation. The detection of a strong local wake is independent evidence that the Magellanic clouds are on their first orbit around the Milky Way. The wake traces the path of the LMC, which will provide insight into the orbit of the LMC, which in turn is a sensitive probe of the mass of the LMC and the Milky Way. These data demonstrate that the outer halo is not in dynamical equilibrium, as is often assumed. The morphology and strength of the wake could be used to test the nature of dark matter and gravity.