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Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I wanted help solving question 12 in the textbook of this linear algebra problem. 

368
CHAPTER 6 Orthogonality and Least Squares
PRACTICE PROBLEMS
1 -3 -3
5
1. Let A =
1
1
and b =
-3
Find a least-squares solution of A:
1
7
2
-5
and compute the associated least-squares error.
2. What can you say about the least-squares solution of Ax = b when b is orthe
to the columns of A?
6.5 EXERCISES
1
2
In Exercises 1–4, find a least-squares solution of Ax = b by
(a) constructing the normal equations for â and (b) solving for âx.
10. A =
-1
4
b =
-1
1
1
1. A =
2 -3
b =
1
4
1
1
11. А —
-1
3
2
-5
1
b
1
2
1
1
-5
2. А —
-2
b =
1
1
2
1
1
-1
12. А —
b =
1
-2
-1
-1
2
b =
3
1
3. А %—
-4
3
2
5
13. Let A =
1
b =
, ar
u =
-1
3
1
3
4. А —D
1
-1
b =
. Compute Au and Av, and compare them
1
1
Could u possibly be a least-squares solution of A
(Answer this without computing a least-squares solut
In Exercises 5 and 6, describe all least-squares solutions of the
equation Ax = b.
2
1
4
1
1
1
14. Let A =
-3
-4
b =
4
u =
ar
1
3
4
3
b =
8
5. A=
1
Compute Au and Av, and compare them wi
1
1
2
it possible that at least one of u or v could be a least-s
solution of Ax = b? (Answer this without computing
1
1
7
1
1
squares solution.)
1
6. А —
1
b
1
3
In Exercises 15 and 16, use the factorization A = QR to
1
1
least-squares solution of Ax = b.
1
4
2/3 -1/3
2/3
2/3
1/3 -2/3
2
3
3
15. А —
2
4
b =
7. Compute the least-squares error associated with the least-
squares solution found in Exercise 3.
1
1
8. Compute the least-squares error associated with the least-
squares solution found in Exercise 4.
1/2 -1/2
1/2
1/2
1/2 -1/2
[2
1
16. А —
1
3
b =
5
-1
1/2
1/2 ]
In Exercises 9-12, find (a) the orthogonal projection of b onto
Col A and (b) a least-squares solution of Ax = b.
In Exercises 17 and 18, A is an m × n matrix and b is in R"
each statement True or False. Justify each answer.
1
4
9. А —
3
1
b =
-2
17. a. The general least-squares problem is to find an
makes Ax as close as possible to b.
-2
4
Transcribed Image Text:368 CHAPTER 6 Orthogonality and Least Squares PRACTICE PROBLEMS 1 -3 -3 5 1. Let A = 1 1 and b = -3 Find a least-squares solution of A: 1 7 2 -5 and compute the associated least-squares error. 2. What can you say about the least-squares solution of Ax = b when b is orthe to the columns of A? 6.5 EXERCISES 1 2 In Exercises 1–4, find a least-squares solution of Ax = b by (a) constructing the normal equations for â and (b) solving for âx. 10. A = -1 4 b = -1 1 1 1. A = 2 -3 b = 1 4 1 1 11. А — -1 3 2 -5 1 b 1 2 1 1 -5 2. А — -2 b = 1 1 2 1 1 -1 12. А — b = 1 -2 -1 -1 2 b = 3 1 3. А %— -4 3 2 5 13. Let A = 1 b = , ar u = -1 3 1 3 4. А —D 1 -1 b = . Compute Au and Av, and compare them 1 1 Could u possibly be a least-squares solution of A (Answer this without computing a least-squares solut In Exercises 5 and 6, describe all least-squares solutions of the equation Ax = b. 2 1 4 1 1 1 14. Let A = -3 -4 b = 4 u = ar 1 3 4 3 b = 8 5. A= 1 Compute Au and Av, and compare them wi 1 1 2 it possible that at least one of u or v could be a least-s solution of Ax = b? (Answer this without computing 1 1 7 1 1 squares solution.) 1 6. А — 1 b 1 3 In Exercises 15 and 16, use the factorization A = QR to 1 1 least-squares solution of Ax = b. 1 4 2/3 -1/3 2/3 2/3 1/3 -2/3 2 3 3 15. А — 2 4 b = 7. Compute the least-squares error associated with the least- squares solution found in Exercise 3. 1 1 8. Compute the least-squares error associated with the least- squares solution found in Exercise 4. 1/2 -1/2 1/2 1/2 1/2 -1/2 [2 1 16. А — 1 3 b = 5 -1 1/2 1/2 ] In Exercises 9-12, find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of Ax = b. In Exercises 17 and 18, A is an m × n matrix and b is in R" each statement True or False. Justify each answer. 1 4 9. А — 3 1 b = -2 17. a. The general least-squares problem is to find an makes Ax as close as possible to b. -2 4
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