John knows that the least squares solution to Ax = b can be identified with the closest point on the subspace mg A spanned by the columns of the coefficient matrix. Therefore, he tries to find the solution by first orthonormalizing the columns using Gram- Schmidt, and then finding the least squares coefficients by the orthonormal basis formula (5.63). To his surprise, he does not get the same solution! Can you explain the source of his difficulty. How can you use his solution to obtain the proper least squares solution x? Check your algorithm with the system that we treated in Example 4.9.