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We introduce a way to learn to estimate a scene representation from a single image by predicting a low-dimensional subspace of optical flow for each training example, which encompasses the variety of possible camera and object movement. Supervision is provided by a novel loss which measures the distance between this predicted flow subspace and an observed optical flow. This provides a new approach to learning scene representation tasks, such as monocular depth prediction or instance segmentation, in an unsupervised fashion using in-the-wild input videos without requiring camera poses, intrinsics, or an explicit multi-view stereo step. We evaluate our method in multiple settings, including an indoor depth prediction task where it achieves comparable performance to recent methods trained with more supervision. Our project page is at https://dimensions-of-motion.github.io/.