A simple modification of the Bartels-Golub update [cf. \textit{R. H. Bartels} and \textit{G. H. Golub}, Commun. ACM 12, 266-268 (1969; Zbl 0181.191)] of the triangular factorization \(PB=LU\) of a given nonsingular matrix B is presented if a column of B is replaced, such that the true new factors L and U are yielded instead of their representations in product form. Finally, the mathematical equivalence to the updating techniques proposed by \textit{R. Fletcher} and \textit{S. P. J. Matthews} [Math. Program. 30, 267-284 (1984; Zbl 0574.65057)] as well as by \textit{J. J. H. Forrest} and \textit{J. A. Tomlin} [ibid. 2, 263-278 (1972; Zbl 0288.90048)] is established thus providing a common ground for the three major techniques of linear programming.