The basis of a tangent space at a point on a differentiable manifold is a set of vectors that spans the tangent space, allowing for the representation of any tangent vector at that point as a linear combination of the basis vectors. This concept is fundamental in differential geometry, providing a local linear approximation of the manifold and facilitating the study of vector fields and differential forms.
Matrix manifolds are smooth surfaces composed of matrices that allow for the application of differential geometry to problems in linear algebra and optimization. They provide a framework for understanding the geometry of matrix spaces, enabling efficient algorithms for a variety of applications including machine learning, computer vision, and control theory.