Tangent Space
=============

Definition
----------

For :math:`x \in \mathcal{M}`, the tangent space :math:`T_x\mathcal{M}` of a manifold :math:`\mathcal{M}` at point :math:`x` is the first order linear approximation of :math:`\mathcal{M}` around :math:`x`.

A tangent space is the "flat" space that approximates the manifold at a specific point, where we can do all the familiar operations of linear algebra, like addition and scalar multiplication.