Foundations of Numerical Linear Algebra

Many computational problems in science, engineering, and data analysis rely on efficient linear algebra. This course demystifies the numerical methods that power these solutions, providing the critical insights needed to understand how computers solve complex mathematical problems. By the end of this course, you will possess the foundational knowledge to understand, implement, and analyze numerical algorithms for solving linear systems, eigenvalue problems, and matrix decompositions, enabling you to confidently approach complex computational challenges. What you'll learn: * Understand fundamental concepts of vectors, matrices, and linear transformations. * Apply direct methods like LU decomposition to solve systems of linear equations. * Explore iterative techniques such as Jacobi and Gauss-Seidel for large-scale problems. * Analyze numerical stability, error propagation, and conditioning in linear algebra algorithms. * Grasp the basics of eigenvalue problems and their numerical solutions. * Learn the conceptual importance and applications of Singular Value Decomposition (SVD). * Practice evaluating the efficiency and accuracy of various numerical methods. The course begins with a review of core linear algebra concepts, then progressively introduces direct and iterative numerical methods for solving linear systems. It covers the intricacies of error analysis, delves into eigenvalue problems, and concludes with an introduction to advanced decomposition techniques. This course is designed for beginners with a basic understanding of mathematics, including algebra and calculus, who are interested in scientific computing, data science, engineering, or machine learning. No prior experience with numerical methods or advanced linear algebra is required. Start building your expertise in the computational backbone of modern data-driven fields today.