Lecture 2: Linear systems and LU (Part II) | Highlights and Annotations by Gistr.

Gaussian elimination solves linear systems (Ax=B). It uses forward substitution to create an upper triangular matrix (U), then back substitution to solve for x. For multiple systems with the same A but different B's, LU factorization (A=LU) is more efficient: factor A into L (lower triangular) and U (upper triangular) once (O(n³)), then solve each system with L and U (O(n²)). Pivoting (row/column swapping) improves numerical stability by prioritizing larger pivots. This segment provides a visual and conceptual explanation of Gaussian elimination's forward substitution phase. The presenter uses a generalized matrix with 'x's to illustrate the process of eliminating variables and achieving an upper triangular matrix. The explanation of forward substitution is clear and concise. This segment demonstrates Gaussian elimination with a 2x2 matrix, showing the steps to eliminate a variable and the resulting matrix. The presenter's questioning approach encourages active learning and clarifies the process.This segment explores the reverse process of Gaussian elimination, transforming the resultant matrix back to the original form. It highlights the systematic approach needed to solve systems of equations and introduces the concept of algorithms. This segment focuses on the back substitution phase of Gaussian elimination, explaining how to solve for variables using the upper triangular matrix obtained in the forward substitution phase. The presenter clearly explains the process and its efficiency. This segment addresses the challenges of zero pivots and numerical instability in Gaussian elimination. It introduces the concept of pivoting as a solution to these problems, highlighting the importance of choosing appropriate pivots to avoid division by small numbers and maintain numerical stability. This segment explains how LU factorization, decomposing a matrix into lower (L) and upper (U) triangular matrices, significantly speeds up solving linear systems of equations. It details the two-step process (solving Ly=B and then Ux=y), highlighting the reduction from n³ to n² time complexity for each step after the initial factorization. The discussion also covers practical considerations like storage optimization by storing L and U within the same matrix space and cautions against treating the combined LU matrix as a standard matrix for vector multiplication.