Elimination with weighted row combinations for solving linear equations and least squares problems

Let A be a matrix of n rows and m columns, m≦n. If and only if the columns are linearly independent, then for any vector b there exists a unique vector x minimizing the Euclidean norm of \(b - Ax,\parallel b - Ax\parallel = \mathop {\min }\limits_\xi \parallel b - A\xi \parallel .\).