Lectures related to Tangent spaces: Linearization of Embedded Submanifolds

Covers linear systems, vector subspaces, and the kernel and image of linear applications.

Explores smooth manifolds through diffeomorphisms and embedded submanifolds in a linear space.

Introduces tangent spaces to submanifolds and their properties in optimization on manifolds.

Tangent vectors without embedding space: Revisiting the embedded case

Explores defining tangent vectors without an embedding space, focusing on creating tangent spaces at every point of a manifold through equivalence classes of curves.

Linear Algebra Basics

Covers the basics of linear algebra, including linear maps, bases, and matrix operations.

Kernel, Image and Linear Maps

Explains kernel, image, and linear maps, illustrating concepts with examples.

Linear Algebra: Subspaces and Transformations

Explores subspaces in linear algebra and transformations, including kernels and images of linear transformations.

Differential Calculus: Applications and Reminders

Covers differential calculus applications and reminders, emphasizing the importance of differentiability in mathematical analysis.

Linear Applications: Properties and Examples

Explores properties of linear applications, including symmetric matrices and linearity in analysis.

Linear Independence and Basis

Explains linear independence, basis, and matrix rank with examples and exercises.

Gradients on Riemannian submanifolds, local frames

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

Linear Transformations: Matrices and Applications

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

Submanifolds: Locally Deformable into Linear Patches

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

Linear Applications: Kernel

Introduces the kernel of a linear application and its properties.

Group Actions: Differential of Orbit Map

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

Linear Applications: Matrices and Spaces

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

Linear Mapping Basics

Covers the basics of linear mapping and coordinate systems.

Linear Algebra: Rank Theorem

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

Linear Independence: Definition and Examples

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

Vector Spaces and Linear Applications

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