Adams, G.Griffin, M. F.Stewart, G. W.Appeared in Proceedings of ACASSP-91. An algorithm for updating the null space of a matrix is described. The algorithm is based on a new decomposition, called the URV decomposition, which can be updated in $O(N^2)$ and serves as an intermediary between the QR decomposition and the singular value decomposition. The URV decomposition is applied to a high-resolution direction of arrival problem based on the MUSIC algorithm. A virtue of the updating algorithm is the running estimate of rank. Additional files are available via anonymous ftp at: thales.cs.umd.edu in the directory pub/reports (Also cross-referenced as UMIACS-TR-91-46)en-USTechnical Report