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In this note, following suggestions by Tao, we extend the randomizedalgorithm for linear equations over prime fields by Raghavendra to a randomizedalgorithm for linear equations over the reals. We also show that the algorithmcan be parallelized to solve a system of linear equations $A x = b$ with aregular $n \times n$ matrix $A$ in time $O(n^2)$, with probability one. Notethat we do not assume that $A$ is symmetric.

A Randomized Parallel Algorithm with Run Time $O(n^2)$ for Solving an $n \times n$ System of Linear Equations