Orthogonal Matrices and Least Squares Method

This lecture covers the concept of orthogonal matrices, stating that the columns of a matrix U are orthonormal if and only if UTU equals the identity matrix. It also discusses the projection onto a subspace, the properties of orthogonal matrices, and the method of least squares for finding the best-fitting line through given points in a plane. The lecture further explores the application of orthogonal matrices in linear transformations and the theorem of best approximation in vector spaces. Additionally, it explains the concept of least squares solutions for the equation Ax=b, emphasizing the importance of the normal equation ATA=ATb. The instructor provides geometric interpretations and practical implications of these mathematical concepts.