Find a QR factorization of the matrix in Exercise 12.
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 16 in the textbook for
![6.4 The Gram-
7. Find an orthonormal basis of the subspace spanned by the
vectors in Exercise 3.
QR, where Q is m
19. Suppose A =
that if the columns of A are linearly
be invertible. [Hint: Study the eqr
8. Find an orthonormal basis of the subspace spanned by the
vectors in Exercise 4.
fact that A =
= QR.]
20. Suppose A = QR, where R is a
that A and Q have the same colum
Col A, show that y = Qx for some
show that y = Ax for some x.]
Find an orthogonal basis for the column space of each matrix in
Exercises 9–12.
3
-5
1
-1
6.
6
1
1
1
3
3
QR as in Theorem 1
orthogonal m × m (square) matrix
upper triangular matrix R such th:
9.
10.
21. Given A =
-1
-2
1
-2
3 -7
8
1
-4
-3
1
2
5
1
5
R
A = 0
e[]
1
1
-4
-1
-3
1
11.
4
-3
12.
2
3
qr command
ization when rank A = n.
1
-4
7
1
2
The MATLAB
supp
1
2
1
1
5
8
22. Let uj,..., u, be an orthogonal
R", and let T : R" → R" be de
In Exercises 13 and 14, the columns of Q were obtained by
applying the Gram-Schmidt process to the columns of A. Find an
upper triangular matrix R such that A =
Show that T is a linear transforma
QR. Check your work.
5/6 -1/6]
5/6
1/6
-3/6
1/6
1/6
3/6
23. Suppose A = QR is a QR facto
trix A (with linearly independen
[A A2], where Aj has
QR factorization of A1, and expl=
has the appropriate properties.
9
1
7
13. А —
colum
-3
-5
1
5
-2
3
-2/7
5/7
24. [M] Use the Gram-Schmidt prc
5/7
2/7
2/7 -4/7
4/7
7
14. A =
produce an orthogonal basis for th
2 -2
4
6
2/7
-10
13
7 -111
1
-5
3
15. Find a QR factorization of the matrix in Exercise 11.
A =
-6
3
13
-3
16. Find a QR factorization of the matrix in Exercise 12.
16 -16
-2
5
1
-5
-7
In Exercises 17 and 18, all vectors and subspaces are in R". Mark
each statement True or False. Justify each answer.
25. [M] Use the method in this sectic
ization of the matrix in Exercise 2
17. a. If {v1, V2, V3} is an orthogonal basis for W, then mul-
26. [M] For a matrix program, the Gra
tiplying v3 by a scalar c gives a new orthogonal basis
{V1, V2, CV3}.
better with orthonormal vectors.
b. The Gram-Schmidt process produces from a linearly in-
dependent set {x , ..., X,} an orthogonal set {v1, . .., Vp}
with the property that for each k, the vectors vị, .… . . , Vk
span the same subspace as that spanned by x1,..., X .
c. If A = QR, where Q has orthonormal columns, then
= Q"A.
in Theorem 11, let A = [ x|
n x k matrix whose columns forr
the subspace Wg spanned by the
for x in R", QQ"x is the orthogor
(Theorem 10 in Section 6.3). If x
then equation (2) in the proof of T
R =
Vk+1 = Xk+1 – Q(Q*xx+1)
18. a. If W = Span {x1, X2, X3} with {x1, X2, X3} linearly inde-
(The parentheses above reduce
operations.) Let uk+1
next step is [ Q u+1]. Use this
QR factorization of the matrix
keystrokes or commands you use.
pendent, and if {V1, V2, V3} is an orthogonal set in W , then
{V1, V2, V3} is a basis for W.
Vk+1/|
b. If x is not in a subspace W, then x – projw x is not zero.
c. In a QR factorization, say A = QR (when A has lin-
early independent columns), the columns of Q form an
orthonormal basis for the column space of A.
WEB
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Transcribed Image Text:6.4 The Gram-
7. Find an orthonormal basis of the subspace spanned by the
vectors in Exercise 3.
QR, where Q is m
19. Suppose A =
that if the columns of A are linearly
be invertible. [Hint: Study the eqr
8. Find an orthonormal basis of the subspace spanned by the
vectors in Exercise 4.
fact that A =
= QR.]
20. Suppose A = QR, where R is a
that A and Q have the same colum
Col A, show that y = Qx for some
show that y = Ax for some x.]
Find an orthogonal basis for the column space of each matrix in
Exercises 9–12.
3
-5
1
-1
6.
6
1
1
1
3
3
QR as in Theorem 1
orthogonal m × m (square) matrix
upper triangular matrix R such th:
9.
10.
21. Given A =
-1
-2
1
-2
3 -7
8
1
-4
-3
1
2
5
1
5
R
A = 0
e[]
1
1
-4
-1
-3
1
11.
4
-3
12.
2
3
qr command
ization when rank A = n.
1
-4
7
1
2
The MATLAB
supp
1
2
1
1
5
8
22. Let uj,..., u, be an orthogonal
R", and let T : R" → R" be de
In Exercises 13 and 14, the columns of Q were obtained by
applying the Gram-Schmidt process to the columns of A. Find an
upper triangular matrix R such that A =
Show that T is a linear transforma
QR. Check your work.
5/6 -1/6]
5/6
1/6
-3/6
1/6
1/6
3/6
23. Suppose A = QR is a QR facto
trix A (with linearly independen
[A A2], where Aj has
QR factorization of A1, and expl=
has the appropriate properties.
9
1
7
13. А —
colum
-3
-5
1
5
-2
3
-2/7
5/7
24. [M] Use the Gram-Schmidt prc
5/7
2/7
2/7 -4/7
4/7
7
14. A =
produce an orthogonal basis for th
2 -2
4
6
2/7
-10
13
7 -111
1
-5
3
15. Find a QR factorization of the matrix in Exercise 11.
A =
-6
3
13
-3
16. Find a QR factorization of the matrix in Exercise 12.
16 -16
-2
5
1
-5
-7
In Exercises 17 and 18, all vectors and subspaces are in R". Mark
each statement True or False. Justify each answer.
25. [M] Use the method in this sectic
ization of the matrix in Exercise 2
17. a. If {v1, V2, V3} is an orthogonal basis for W, then mul-
26. [M] For a matrix program, the Gra
tiplying v3 by a scalar c gives a new orthogonal basis
{V1, V2, CV3}.
better with orthonormal vectors.
b. The Gram-Schmidt process produces from a linearly in-
dependent set {x , ..., X,} an orthogonal set {v1, . .., Vp}
with the property that for each k, the vectors vị, .… . . , Vk
span the same subspace as that spanned by x1,..., X .
c. If A = QR, where Q has orthonormal columns, then
= Q"A.
in Theorem 11, let A = [ x|
n x k matrix whose columns forr
the subspace Wg spanned by the
for x in R", QQ"x is the orthogor
(Theorem 10 in Section 6.3). If x
then equation (2) in the proof of T
R =
Vk+1 = Xk+1 – Q(Q*xx+1)
18. a. If W = Span {x1, X2, X3} with {x1, X2, X3} linearly inde-
(The parentheses above reduce
operations.) Let uk+1
next step is [ Q u+1]. Use this
QR factorization of the matrix
keystrokes or commands you use.
pendent, and if {V1, V2, V3} is an orthogonal set in W , then
{V1, V2, V3} is a basis for W.
Vk+1/|
b. If x is not in a subspace W, then x – projw x is not zero.
c. In a QR factorization, say A = QR (when A has lin-
early independent columns), the columns of Q form an
orthonormal basis for the column space of A.
WEB
3.
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