Tangent vectors without embedding space: Making tangent spaces linear

This lecture covers the concept of tangent vectors without an embedding space, focusing on making tangent spaces linear. The tangent space TyM is defined as the quotient set Cx/~, where a tangent vector v in TM is an equivalence class of curves. The lecture explores how to 'add' and 'scale' two tangent vectors, as well as the injectivity and surjectivity of the tangent space. It also discusses the independence of the equivalence relation on Cx with respect to the choice of chart around x. If M is embedded, two notions of tangent spaces are presented, showing their equivalence and the linear bijection from Cx/~ to ker Dh(x).