MATLAB EXERCISE: Let A = [6 3 6 1, 1 2 1 6, 3 6 3 2, 6 3 6 1, 2 1 2 3, 3 6 3 2, 2 1 2 3, 1 2 1 6 ] . Show all work in Matlab Code.

(a) Show that the columns of A are orthogonal by making an appropriate matrix calculation.

(b) Form the matrix U by making the columns of A orthonormal.

(c) Compute UTU and UUT. How do they differ? Explain why.

(d) Generate a random vector y ? R 8 , and compute w = UUTy and z = y - w. Verify that z is orthogonal to w.

(e) Verify that z is orthogonal to each column of U. (f) Notice that y = w + z with w ? Col A. Explain why z is in Col A?.