👉 Linear Algebra (LU) is a method used to solve systems of linear equations by decomposing the coefficient matrix into a lower triangular matrix (L) and an upper triangular matrix (U). This decomposition leverages the properties of matrices to transform the original system \(Ax = b\) into two simpler triangular systems: \(Ly = b\) and \(Ux = y\), which can be solved more efficiently than the original system. The LU decomposition is particularly useful in numerical computations due to its efficiency and stability, especially when combined with techniques like LU factorization with partial pivoting. This method is foundational in many areas of applied mathematics, computer science, and engineering, including solving linear regression problems, finite element analysis, and optimization tasks.
