The Row picture of least squares and symmedian coordinates

In this work, we study the least squares problem for an overdetermined system Ax ¼ b, where A 2 Rmn with m[ n, which finds the vector x minimizing the Euclidean norm of the residual b Ax. While the classical column picture provides a geometric interpretation in Rm, the row picture offers an interpretation in Rn. We show that, for a 3 2 system, the least squares solution lies within the corresponding triangle, and for an ðn þ 1Þ n system, within the corresponding simplex. In the general case, the solution resides in the closed and bounded region defined by the system’s hyperplanes. We further relate the least squares solution to the symmedian point in the simplex case and introduce symmedian coordinates, demonstrating their connection to trilinear and barycentric coordinates. These results establish explicit relationships between least squares solutions and classical coordinate systems, providing geometric insight into the structure of the solution.