LU Decomposition: Existence

This lecture covers the LU decomposition of a matrix A, where A is decomposed into the product of a lower triangular matrix L and an upper triangular matrix U. The process involves performing operations on the columns of A to obtain L and U. The lecture also discusses reducing A to an echelon form matrix using type I and II operations, leading to the LU decomposition. The existence of L and U matrices is proven through examples and propositions, demonstrating the usefulness of LU decomposition in matrix operations.