Page 322 - Matrix Analysis & Applied Linear Algebra

318 Chapter 5 Norms, Inner Products, and Orthogonality

5.5.2. Use the Gram–Schmidt procedure to find an orthonormal basis for the

1 −2 3 −1
four fundamental subspaces of A = 2 −4 6 −2 .
3 −6 9 −3

5.5.3. Apply the Gram–Schmidt procedure with the standard inner product
!
i 0 0
3
for C to i , i , 0 .
i i i
5.5.4. Explain what happens when the Gram–Schmidt process is applied to an
orthonormal set of vectors.

5.5.5. Explain what happens when the Gram–Schmidt process is applied to a
linearly dependent set of vectors.

   
1 0 −1 1
1 2 1 1
5.5.6. Let A =   and b =   .
1 1 −3 1
0 1 1 1
(a) Determine the rectangular QR factorization of A.
(b) Use the QR factors from part (a) to determine the least squares
solution to Ax = b.

5.5.7. Given a linearly independent set of vectors S = {x 1 , x 2 ,..., x n } in an
inner-product space, let S k = span {x 1 , x 2 ,..., x k } for k =1, 2,...,n.
Give an induction argument to prove that if O k = {u 1 , u 2 ,..., u k } is
the Gram–Schmidt sequence defined in (5.5.2), then O k is indeed an or-
thonormal basis for S k = span {x 1 , x 2 ,..., x k } for each k =1, 2,...,n.

5.5.8. Prove that if rank (A m×n )= n, then the rectangular QR factorization
of A is unique. That is, if A = QR, where Q m×n has orthonormal
columns and R n×n is upper triangular with positive diagonal entries,
then Q and R are unique. Hint: Recall Example 3.10.7, p. 154.

5.5.9. (a) Apply classical Gram–Schmidt with 3-digit floating-point arith-
!
1 1 1
metic to x 1 = 0 , x 2 = 0 , x 3 = 10 −3 . You may
10 −3 0 0
√
assume that fl 2 =1.41.
(b) Again using 3-digit floating-point arithmetic, apply the modified
Gram–Schmidt algorithm to {x 1 , x 2 , x 3 } , and compare the re-
sult with that of part (a).