Office hours notes: Many students made similar errors with absolute values and error calculations. Inaccurate inequality use was also common. Students should email for help outside of office hours. Lecture covered Cholesky factorization, sparse matrices (harmonic parameterization example), and the ill-conditioning of AᵀA in least squares problems. QR factorization will address this. This segment highlights recurring mistakes students make when dealing with absolute values and inequalities, emphasizing the importance of careful handling of positive and negative numbers within equations. The professor notes at least nine students made the same errors. This segment underscores the significance of understanding the specific characteristics of a numerical problem (e.g., symmetry, positive definiteness) to optimize the solution method. The professor emphasizes that deeper problem knowledge leads to better algorithm design.This segment introduces harmonic parameterization, a technique used in computer graphics to map 3D models onto 2D texture planes for efficient rendering. The professor explains the problem and the classical approach to solving it. The instructor addresses common student questions and concerns regarding homework assignments and office hours. They emphasize the importance of emailing questions in advance rather than during office hours to allow for adequate preparation and avoid disruptions. The next homework assignment is also announced. This segment discusses the challenges of storing and manipulating sparse matrices, which contain a large number of zero elements. The professor explains how Gaussian elimination can destroy sparsity, leading to increased computational cost and storage requirements. The professor also mentions the software SuiteSparse. This segment explains how column operations, unlike row operations, do not affect the column space of a matrix. This understanding is crucial for solving least squares problems more efficiently by transforming the matrix into an orthogonal matrix (Q) and an upper triangular matrix (R) through QR factorization, simplifying the solution process significantly. This segment provides a geometric interpretation of vector projection, demonstrating how to project one vector onto another. It connects this geometric concept to the least squares problem, showing how the solution to the least squares problem directly yields the projection vector and proving that the projection is parallel to the target vector, while the residual is orthogonal to it.