Tangent vectors without embedding space: Revisiting the embedded case

This lecture focuses on defining tangent vectors to a set without referencing an embedding space, aiming to create tangent spaces at every point of a manifold in a way that they are linear spaces. The instructor explains the concept of tangent vectors using curves on the manifold and their velocities, introducing the notion of equivalence classes of curves passing through a point with the same velocity. By defining an equivalence relation on the set of curves, the lecture demonstrates how to construct the tangent space to the manifold at that point. The process involves abstracting the concrete understanding of tangent spaces from embedded submanifolds to apply it to abstract manifolds, emphasizing the importance of separating concerns between curves on the manifold and their velocities.