I wanted help solving question 12 in the textbook of this linear algebra problem.
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
Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.
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![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
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