A fundamental idea in matrix linear algebra is the factorization of a matrix into simpler matrices, such as orthogonal, tridiagonal, and triangular. In this talk we extend this idea to a continuous setting, asking: "What are thecontinuous analogues of matrix factorizations?" The answer we develop involves functions of two variables, an iterative variant of Gaussian elimination, and sufficient conditions for convergence. This leads to a test for non-negative definite kernels, a continuous definition of a triangular quasimatrix (a matrix whose columns are functions), and a fresh perspective on a classic subject. This is work is with Nick Trefethen.