Orthogonality: Norm, Scalar Product, Perpendicularity

This lecture covers the concepts of norm, scalar product, and perpendicularity in R^n, including the definition of norm, calculation of scalar products, and the geometric explanation of orthogonal vectors. It also explores the notion of perpendicularity in R^2 and R^3, the theorem of Pythagoras, and the concept of orthogonal complements. The lecture delves into the properties of orthogonal complements, the calculation of distances between vectors, and the theorem of Pythagoras in a generalized form. Additionally, it discusses the orthogonal projection, the concept of orthogonal complements in subspaces, and the relationship between the image and kernel of a matrix.