In certain signal processing applications it is required to compute the null space of a matrix whose rows are samples of a signal. The usual tool for doing this is the singular value decomposition. However, the singular value decomposition has the drawback that it requires $O(p^3)$ operations to recompute when a new sample arrives. In this paper, we show that a different decomposition, called the URV, decomposition is equally effective in exhibiting the null space and can be updated in $O(p^2)$ time. The updating technique can be run on a linear array of $p$ processors in $O(p)$ time. (Also cross-referenced as UMIACS-TR-90-86) To appear in IEEE Transactions on Acoustics, Speech and Signal Processing Additional files are available via anonymous ftp at: thales.cs.umd.edu in the directory pub/reports