Linear Algebra: Singular Value Decomposition

This lecture covers advanced topics in linear algebra, focusing on singular value decomposition (SVD). It explains the concept of column space, null space, and orthogonal complement of a matrix. The lecture also introduces the theorem of SVD, which decomposes any real matrix into orthogonal matrices and a diagonal matrix. Additionally, it discusses the spectral theorem for symmetric matrices and properties of idempotent and orthogonal projection matrices. The lecture concludes with the optimal linear dimension reduction theorem, which highlights the importance of projecting random variables onto subspaces spanned by principal components.