Numerical Analysis: Direct Methods for Linear Systems

This lecture focuses on the resolution of linear systems using direct methods in numerical analysis. The instructor begins by discussing the importance of understanding linear systems and their applications in various fields. The lecture covers the definition of linear systems, represented in matrix form as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the result vector. The instructor emphasizes the significance of matrix properties, such as invertibility, and the implications for finding unique solutions. The discussion includes triangular matrices, which simplify the solving process, and the elimination methods, particularly Gaussian elimination. The instructor explains the algorithmic approach to transforming matrices into triangular form, highlighting the steps of normalization and elimination. The lecture also addresses the challenges posed by poorly conditioned systems and the importance of numerical stability. Finally, the instructor introduces the LU decomposition method, which allows for efficient resolution of multiple linear systems, and discusses the Cholesky decomposition for symmetric positive definite matrices.