5. {(1, 2, 3), (1, −1, 2), (1, –4, 1)}.
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
Solve #5, SHow all of your work and give explanations. POST PICTURES OF YOUR WORK
![296 CHAPTER 4
Vector Spaces
• Know what information the Wronskian does (and does
not) give about the linear dependence or linear inde-
pendence of a set of functions on an interval I.
True-False Review
For Questions (a)-(i), 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.
(a) Every vector space V possesses a unique minimal
spanning set.
(b) The set of column vectors of a 5 x 7 matrix A must be
linearly dependent.
(c) The set of column vectors of a 7 x 5 matrix A must be
linearly independent.
(d) Any nonempty subset of a linearly independent set of
vectors is linearly independent.
(e) If the Wronskian of a set of functions is nonzero at
some point xo in an interval I, then the set of func-
tions is linearly independent.
(f) If it is possible to express one of the vectors in a set
S as a linear combination of the others, then S is a
linearly dependent set.
(g) If a set of vectors S in a vector space V contains a
linearly dependent subset, then S is itself a linearly
dependent set.
(h) A set of three vectors in a vector space V is linearly de-
pendent if and only if all three vectors are proportional
to one another.
(i) If the Wronskian of a set of functions is identically
zero at every point of an interval I, then the set of
functions is linearly dependent.
Problems
For Problems 1-10, determine whether the given set of
vectors is linearly independent or linearly dependent in
R". In the case of linear dependence, find a dependency
relationship.
1. {(3, 6, 9)}.
2. {(1, 1), (1, 1)}.
3. {(2, 1), (3, 2), (0, 1)).
4. {(1, 1, 0), (0, 1, −1), (1, 1, 1)}.
5. {(1, 2, 3), (1, −1, 2), (1, −4, 1)}.
6. {(2, 4, 6), (3, -6, 9)}.
7. {(1, 1, 2), (2, 1, 0)).
8. {(1, 1, 2), (0, 2, 1), (3, 1, 2), (-1,-1, 1)}.
9. {(1, 1, 2, 3), (2, −1, 1, −1), (−1, 1, 1, 1)}.
10. {(2, 1, 0, 1), (1, 0, −1, 2), (0, 3, 1, 2),
(-1, 1, 2, 1)).
11. Let v₁ = (1, 2, 3), v₂ = (4, 5, 6), V3 = (7, 8, 9). De-
termine whether {V1, V2, V3} is linearly independent in
R³. Describe
span{V1, V2, V3}
geometrically.
12. Consider the vectors v₁ = (2, -1, 5), v₂ = (1, 3, –4),
V3 = (-3, 9, 12) in R³.
(a) Show that {V₁, V2, V3} is linearly dependent.
(b) Is v₁ € span{v2, v3}? Draw a picture illustrating
your answer.
13. Determine all values of the constant k for which the
vectors (1, 1, k), (0, 2, k) and (1, k, 6) are linearly de-
pendent in R³.
For Problems 14-15, determine all values of the constant k
for which the given set of vectors is linearly independent
in R4.
14. {(1, 0, 1, k), (-1, 0, k, 1), (2, 0, 1, 3)}.
15. {(1, 1, 0, 1), (1, k, 1, 1), (2, 1, k, 1), (−1, 1, 1, k)}.
For Problems 16-18, determine whether the given set of vec-
tors is linearly independent in M₂ (R).
- [6₂1]^² - [6 ]-^² - [84]
,
-
, A3 =
01
16. A₁ =
- [34]^2-[13]
=
17. A₁ =
- [12]·^² = [ 2² ]-^² = [3₂²]
, A₂
, A3
18. A₁ =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2F8e023838-00cd-45d1-a987-566198177d93%2Fp0i09cj_processed.png&w=3840&q=75)
Transcribed Image Text:296 CHAPTER 4
Vector Spaces
• Know what information the Wronskian does (and does
not) give about the linear dependence or linear inde-
pendence of a set of functions on an interval I.
True-False Review
For Questions (a)-(i), 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.
(a) Every vector space V possesses a unique minimal
spanning set.
(b) The set of column vectors of a 5 x 7 matrix A must be
linearly dependent.
(c) The set of column vectors of a 7 x 5 matrix A must be
linearly independent.
(d) Any nonempty subset of a linearly independent set of
vectors is linearly independent.
(e) If the Wronskian of a set of functions is nonzero at
some point xo in an interval I, then the set of func-
tions is linearly independent.
(f) If it is possible to express one of the vectors in a set
S as a linear combination of the others, then S is a
linearly dependent set.
(g) If a set of vectors S in a vector space V contains a
linearly dependent subset, then S is itself a linearly
dependent set.
(h) A set of three vectors in a vector space V is linearly de-
pendent if and only if all three vectors are proportional
to one another.
(i) If the Wronskian of a set of functions is identically
zero at every point of an interval I, then the set of
functions is linearly dependent.
Problems
For Problems 1-10, determine whether the given set of
vectors is linearly independent or linearly dependent in
R". In the case of linear dependence, find a dependency
relationship.
1. {(3, 6, 9)}.
2. {(1, 1), (1, 1)}.
3. {(2, 1), (3, 2), (0, 1)).
4. {(1, 1, 0), (0, 1, −1), (1, 1, 1)}.
5. {(1, 2, 3), (1, −1, 2), (1, −4, 1)}.
6. {(2, 4, 6), (3, -6, 9)}.
7. {(1, 1, 2), (2, 1, 0)).
8. {(1, 1, 2), (0, 2, 1), (3, 1, 2), (-1,-1, 1)}.
9. {(1, 1, 2, 3), (2, −1, 1, −1), (−1, 1, 1, 1)}.
10. {(2, 1, 0, 1), (1, 0, −1, 2), (0, 3, 1, 2),
(-1, 1, 2, 1)).
11. Let v₁ = (1, 2, 3), v₂ = (4, 5, 6), V3 = (7, 8, 9). De-
termine whether {V1, V2, V3} is linearly independent in
R³. Describe
span{V1, V2, V3}
geometrically.
12. Consider the vectors v₁ = (2, -1, 5), v₂ = (1, 3, –4),
V3 = (-3, 9, 12) in R³.
(a) Show that {V₁, V2, V3} is linearly dependent.
(b) Is v₁ € span{v2, v3}? Draw a picture illustrating
your answer.
13. Determine all values of the constant k for which the
vectors (1, 1, k), (0, 2, k) and (1, k, 6) are linearly de-
pendent in R³.
For Problems 14-15, determine all values of the constant k
for which the given set of vectors is linearly independent
in R4.
14. {(1, 0, 1, k), (-1, 0, k, 1), (2, 0, 1, 3)}.
15. {(1, 1, 0, 1), (1, k, 1, 1), (2, 1, k, 1), (−1, 1, 1, k)}.
For Problems 16-18, determine whether the given set of vec-
tors is linearly independent in M₂ (R).
- [6₂1]^² - [6 ]-^² - [84]
,
-
, A3 =
01
16. A₁ =
- [34]^2-[13]
=
17. A₁ =
- [12]·^² = [ 2² ]-^² = [3₂²]
, A₂
, A3
18. A₁ =
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