(a) Show that applying the Gram-Schmidt algorithm to the columns of A produces an orthonormal basis for mg A.
(b) Prove that this is equivalent to the matrix factorization A = Q R, where Q is an m × n matrix with orthonormal columns, while R is a nonsingular n x n upper triangular matrix.
(c) Show that the QR program in the text also works for rectangular, m × n, matrices as stated, the only modification being that the row indices i run from 1 to m.
(d) Apply this method to factor
(i)

(ii)

(iii)

(iv)

(e) Explain what happens if rank A

. 132 211 3-4 1235 1110 2122