Finding the orthogonal projection of a point onto an affine subspace

A simple method is proposed to find the orthogonal projection of a given point to the solution set of a system of linear equations. This is also a direct method for solving systems of linear equations. The output of the method is either the projection or inconsistency of the system. Moreover, in the process also linearly dependent equations are recognized. This paper is constrained for giving theoretical foundations, computational complexity and some numerical experiments with dense matrices although the method allows to employ sparsity. The raw method could not compete with best software packages in solving linear equations for general matrices, but it was competitive in finding projections for matrices with small number of rows relative to the number of columns.