In parts a-d let u be the matrix formed by normalizing each

Question

Question: In parts (a)-(d), let U be the matrix formed by normalizing each column of the matrix A in Exercise.

a. Compute U^TU and U U^T. How do they differ?

b. Generate a random vector y in R⁸, and compute p = U U^Ty and z = y - p. Explain why p is in Col A. Verify that z is orthogonal to p.

c. Verify that z is orthogonal to each column of U.

d. Notice that y = p + z, with p in Col A. Explain why z is in (Col A)⊥?. (The significance of this decomposition of y will be explained in the next section.)

Exercise: Show that the columns of the matrix A are orthogonal by making an appropriate matrix calculation. State the calculation you use.