Suppose we are able to write the matrix A as a product of two matrices, L.U = A, where L is lower triangular (has elements only on the diagonal and below) and U is upper triangular (has elements only on the diagonal and above). We can use a decomposition such as (L.U = A) to sole the linear set A.x = (L.U).x = L.(U.x) = b

by first solving for the vector y such that L.y = b

and then solving U.x = y