Orthogonality and Projection

This lecture introduces the concept of orthogonality, emphasizing the importance of the scalar product and the notion of orthogonal complement. The instructor explains how to determine if two vectors are orthogonal based on their scalar product being zero. The lecture covers the definition of orthogonal bases and the projection of a vector onto a subspace. The process of finding the orthogonal projection is detailed, highlighting the uniqueness of the decomposition and its independence from the chosen basis. The instructor also touches on the Graham-Schmidt process for constructing an orthogonal basis from a non-orthogonal one, showcasing the practical applications of orthogonality in vector spaces.