We present a new algorithm for computing a µ-basis of the syzygy module of n polynomials in one variable over an arbitrary field K. The algorithm is conceptually different from the previously-developed algorithms by Cox, Sederberg, Chen, Zheng, and Wang for n = 3, and by Song and Goldman for an arbitrary n. The algorithm involves computing a "partial" reduced row-echelon form of a (2d+ 1)×n(d+ 1) matrix over K, where d is the maximum degree of the input polynomials. The proof of the algorithm is based on standard linear algebra and is completely self-contained. The proof includes a proof of the existence of the µ-basis and as a consequence provides an alternative proof of the freeness of the syzygy module. The theoretical (worst case asymptotic) computational complexity of the algorithm is O(d 2 n + d 3 + n 2 ). We have implemented this algorithm (HHK) and the one developed by Song and Goldman (SG). Experiments on random inputs indicate that SG is faster than HHK when d is sufficiently large for a fixed n, and that HHK is faster than SG when n is sufficiently large for a fixed d.