Orthogonal Projection: Uniqueness and Properties

This lecture covers the concept of orthogonal projection, focusing on the uniqueness of decomposition and properties. It explains how any vector can be decomposed into orthogonal components, and how the orthogonal projection does not depend on the basis chosen. The lecture also discusses the associated matrix of the orthogonal projection and its properties, such as linearity. Examples are provided to illustrate the application of orthogonal projection, including finding the projection on a plane and best quadratic approximation. The importance of orthonormal bases and the normalization of vectors are highlighted, along with the significance of orthogonal matrices in projection calculations.