Applied Linear Algebra

12.8 Least squares and QR factorization. Suppose A is an m x n matrix with linearly inde- pendent columns and QR factorization A : QR, and b is an m-vector. The vector As? is the linear combination of the columns of A that is closest to the vector b, i.., it is the projection of (1 onto the set of linear combinations of the columns of A. (a) Show that Ail?) : QQTb. (The matrix QQT is called the projection matrix.) (b) Show that ||A bll2 : |lbl|2 llQTbll2. (This is the square of the distance between I) and the closest linear combination of the columns of A.)