Tangent spaces: Linearization of Embedded Submanifolds

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Discusses gradients on Riemannian submanifolds and the construction of local frames.

Covers linear transformations using matrices, focusing on linearity, image, and kernel.

Explores how submanifolds can be deformed into linear patches using diffeomorphisms and the inverse function theorem.

Introduces the kernel of a linear application and its properties.

Explores the differential of group actions on vector spaces and the behavior of orbit maps.

Covers linear applications, matrices, and vector spaces, emphasizing the concept of linear independence.

Covers the basics of linear mapping and coordinate systems.

Covers the Rank Theorem in linear algebra, focusing on vector spaces and linear applications.

Explores the concept of linear independence in vector spaces through definitions and illustrative examples.

Covers vector spaces, subspaces, kernel, image, linear independence, and bases in linear algebra.