What is a (sub)manifold

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Explains linear independence, bases, and dimension in vector spaces, including the importance of the order of vectors in a basis.

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

Covers matrix kernels, images, linear applications, independence, and bases in vector spaces.

Covers the properties and operations of vector spaces, including addition and scalar multiplication.

Explores bases, dimensions, and matrix ranks in vector spaces with practical examples and proofs.

Covers the concept of lines spaces and equivalent matrices in linear algebra.

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

Explores matrix operations, linear systems, solutions, and the span of vectors in linear algebra.

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

Explores orthogonality between vectors and subspaces, demonstrating practical implications in matrix operations.