Null space theory predicts that a neuron will often generate spikes not to produce behavior, but to prevent another neuron's impact on behavior. Here, we present a direct test of this theory in the brain. In the marmoset cerebellum, spike-triggered averaging identified a vector for each Purkinje cell (P-cell) along which its spikes displaced the eyes. Two spikes in two different P-cells produced superposition of their vectors. In the resulting population activity, the spikes were canceled if their contributions were perpendicular to the intended movement. Mossy fibers provided a copy of the motor commands and the sensory goal of the movement. Molecular layer interneurons transformed these inputs so that the P-cell population predicted when the movement had reached the goal and should be stopped.