Orthogonal Projection: Spectral Decomposition

This lecture introduces the concept of orthogonal projection and spectral decomposition in vector spaces. It covers the unique representation of a vector as the sum of orthogonal projections onto subspaces, the Gram-Schmidt orthogonalization process, and the best approximation of a vector in a subspace. The instructor explains the decomposition of vectors into orthogonal components, the calculation of orthogonal bases, and the application of orthogonal projection in matrix transformations. The lecture concludes with the factorization QR theorem for matrices with linearly independent columns.