Orthogonal Projections: Gram-Schmidt Method

This lecture covers the concept of orthogonal projections, focusing on the Gram-Schmidt method to construct an orthogonal basis. The instructor explains the importance of following a specific order when orthogonalizing vectors and emphasizes the significance of maintaining the given order. The lecture delves into the process of constructing an orthogonal base using the Gram-Schmidt method, highlighting the steps involved in normalizing vectors and obtaining an orthonormal basis. Additionally, the lecture introduces the QR factorization technique, showcasing its application in finding eigenvalues and eigenvectors of a matrix. The instructor provides insights into the historical significance of the method, drawing parallels to Gauss's contributions in predicting celestial events. The lecture concludes with a discussion on linear regression and the practical implications of QR factorization in solving systems of equations.