For Problems 11-14, determine orthogonal bases for rowspace(A) and colspace(A). 11. A = 12. A = 13. A = [4 1 -3 20 49 -1 1 15 24 33 42 51 3 14 1 -2 1 1 52 -2].
For Problems 11-14, determine orthogonal bases for rowspace(A) and colspace(A). 11. A = 12. A = 13. A = [4 1 -3 20 49 -1 1 15 24 33 42 51 3 14 1 -2 1 1 52 -2].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Please solve #13, Post pictures of your work and show every step of you work. Thank you!
![basis for V.
True-False Review
For Questions (a)–(f), decide if the given statement is true
or false, and give a brief justification for your answer. If true,
you can quote a relevant definition or theorem from the text.
If false, provide an example, illustration, or brief explanation
of why the statement is false.
...9
(a) If {x₁, X2, . Xn} is already an orthogonal basis for
an inner product space V, then applying the Gram-
Schmidt process to this set results in this same set of
vectors.
(b) If x₁ and x2 are orthogonal, then applying the Gram-
Schmidt Process to the set {x₁, X₁ + X2} will result in
the orthogonal set {x1, x2}.
(c) The Gram-Schmidt process applied to the vectors
{X1, X2, X3} yields the same basis as the Gram-Schmidt
process applied to the vectors {X3, X2, X₁}.
(d) The Gram-Schmidt process can only be applied to a
linearly independent set of vectors.
(e) If B₁
{X1, X2} and B₂ {y1, y2} are two bases
for an inner product V such that the Gram-Schmidt
process applied to each of them results in the same
orthogonal basis {V₁, V₂} for V, then B₁ = B₂.
=
=
=
(f) If the Gram-Schmidt Process applied to B1 = {X1, X₂}
yields the orthogonal set {V₁, V2}, then applying the
Gram-Schmidt Process to B2 = {2x₁, 2x2} will yield
the orthogonal set {2v₁, 2v₂}.
5. {(2, 0, 1), (-3, 1, 1), (1, -3, 8)}.
6. {(1, 1, 1, 1), (1, 2, 1, 2)}.
7. {(1, 0, −1, 0), (1, 1, −1, 0), (−1, 1, 0, 1)}
8. {(1, 2, 0, 1), (2, 1, 1, 0), (1, 0, 2, 1)}.
9. {(1, 1, 1, 0), (–1, 0, 1, 1), (2,−1, 2, 1)).
10. {(1, 2, 3, 4, 5), (-7, 0, 1, -2, 0)}.
For Problems 11-14, determine orthogonal bases for
rowspace(A) and colspace(A).
11. A
12. A
13. A
=
=
-3 20
4 -9 -1 1
14. A =
15
24
33
4 2
5 1
-C
3 14
-2 1
52
1
-2
-4
6-8
0 5
2]
For Problems 15-16, determine an orthonormal basis for the
subspace of C³ spanned by the given set of vectors. Make
sure that you use the appropriate inner product in C3](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2Fa0939d23-2a94-4e7d-af15-c314721a73b2%2F1ut7dl5_processed.png&w=3840&q=75)
Transcribed Image Text:basis for V.
True-False Review
For Questions (a)–(f), decide if the given statement is true
or false, and give a brief justification for your answer. If true,
you can quote a relevant definition or theorem from the text.
If false, provide an example, illustration, or brief explanation
of why the statement is false.
...9
(a) If {x₁, X2, . Xn} is already an orthogonal basis for
an inner product space V, then applying the Gram-
Schmidt process to this set results in this same set of
vectors.
(b) If x₁ and x2 are orthogonal, then applying the Gram-
Schmidt Process to the set {x₁, X₁ + X2} will result in
the orthogonal set {x1, x2}.
(c) The Gram-Schmidt process applied to the vectors
{X1, X2, X3} yields the same basis as the Gram-Schmidt
process applied to the vectors {X3, X2, X₁}.
(d) The Gram-Schmidt process can only be applied to a
linearly independent set of vectors.
(e) If B₁
{X1, X2} and B₂ {y1, y2} are two bases
for an inner product V such that the Gram-Schmidt
process applied to each of them results in the same
orthogonal basis {V₁, V₂} for V, then B₁ = B₂.
=
=
=
(f) If the Gram-Schmidt Process applied to B1 = {X1, X₂}
yields the orthogonal set {V₁, V2}, then applying the
Gram-Schmidt Process to B2 = {2x₁, 2x2} will yield
the orthogonal set {2v₁, 2v₂}.
5. {(2, 0, 1), (-3, 1, 1), (1, -3, 8)}.
6. {(1, 1, 1, 1), (1, 2, 1, 2)}.
7. {(1, 0, −1, 0), (1, 1, −1, 0), (−1, 1, 0, 1)}
8. {(1, 2, 0, 1), (2, 1, 1, 0), (1, 0, 2, 1)}.
9. {(1, 1, 1, 0), (–1, 0, 1, 1), (2,−1, 2, 1)).
10. {(1, 2, 3, 4, 5), (-7, 0, 1, -2, 0)}.
For Problems 11-14, determine orthogonal bases for
rowspace(A) and colspace(A).
11. A
12. A
13. A
=
=
-3 20
4 -9 -1 1
14. A =
15
24
33
4 2
5 1
-C
3 14
-2 1
52
1
-2
-4
6-8
0 5
2]
For Problems 15-16, determine an orthonormal basis for the
subspace of C³ spanned by the given set of vectors. Make
sure that you use the appropriate inner product in C3
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