the ear nt. is lie he in ns ne Each statement in Exercises 33-38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) 33. If v₁...., V4 are in R4 and v3 = 2v₁ + V2, then {V₁, V2, V3, V4) is linearly dependent. 34. If V₁..... V are in R¹ and v3 = 0, then {V₁, V2, V3, V4) is linearly dependent. 35. If v₁ and v₂ are in R4 and v₂ is not a scalar multiple of VI then {V₁, V₂} is linearly independent. 36. If v₁...., V4 are in R* and v3 is not a linear combination of V₁, V2, V4, then {V₁, V2, V3, V4} is linearly independent. 37. If v₁...., V4 are in R4 and {V₁, V2, V3} is linearly dependent. then {V₁, V2, V3. V4) is also linearly dependent. 38. If V₁..... V4 are linearly independent vectors in R¹, then (V1, V2. V3) is also linearly independent. [Hint: Think about X₁V₁ + X₂V₂ + x3V3 +0 V4 = 0.]
the ear nt. is lie he in ns ne Each statement in Exercises 33-38 is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is true, give a justification. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.) 33. If v₁...., V4 are in R4 and v3 = 2v₁ + V2, then {V₁, V2, V3, V4) is linearly dependent. 34. If V₁..... V are in R¹ and v3 = 0, then {V₁, V2, V3, V4) is linearly dependent. 35. If v₁ and v₂ are in R4 and v₂ is not a scalar multiple of VI then {V₁, V₂} is linearly independent. 36. If v₁...., V4 are in R* and v3 is not a linear combination of V₁, V2, V4, then {V₁, V2, V3, V4} is linearly independent. 37. If v₁...., V4 are in R4 and {V₁, V2, V3} is linearly dependent. then {V₁, V2, V3. V4) is also linearly dependent. 38. If V₁..... V4 are linearly independent vectors in R¹, then (V1, V2. V3) is also linearly independent. [Hint: Think about X₁V₁ + X₂V₂ + x3V3 +0 V4 = 0.]
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
33,37
![62 CHAPTER 1 Linear Equations in Linear Algebra
9. V₁ =
10. V₁ =
11.
13.
-[-]--
-3
2
15.
In Exercises 11-14, find the value(s) of h for which the vectors
are linearly dependent. Justify each answer.
17.
-----
=
10
-3
-
-
-5
-
3
5
3
-5
7
-8
19. 12
-4
-5
BED AND
5
-9
7
6
8
5
h
1.[8]
0
9
6
-6
5
4
3
9
Determine by inspection whether the vectors in Exercises 15-20
are linearly independent. Justify each answer.
[{][B][B][-]
1
12.
14.
16.
2
-9
h
2
HO
-4
7
-3
18.
6
[][]
-2
9
2
20.
4 0₁
In Exercises 21 and 22, mark each statement True or False. Justify
each answer on the basis of a careful reading of the text.
h
[4][3][³][8]
5
-7
-
-2
5
3
0
[8]
21. a. The columns of a matrix A are linearly independent if the
equation Ax = 0 has the trivial solution.
b. If S is a linearly dependent set, then each vector is a linear
combination of the other vectors in S.
c. The columns of any 4 x 5 matrix are linearly dependent.
d. If x and y are linearly independent, and if {x, y, z) is
linearly dependent, then z is in Span {x,y}.
22. /a. Two vectors are linearly dependent if and only if they lie
on a line through the origin.
b. If a set contains fewer vectors than there are entries in the
vectors, then the set is linearly independent.
c. If x and y are linearly independent, and if z is in
Span {x, y), then {x, y, z) is linearly dependent.
d. If a set in R" is linearly dependent, then the set contains
more vectors than there are entries in each vector.
In Exercises 23-26, describe the possible echelon forms of the
matrix. Use the notation of Example 1 in Section 1.2.
23. A is a 3 x 3 matrix with linearly independent columns.
24. A is a 2 x 2 matrix with linearly dependent columns,
25. A is a 4 × 2 matrix, A = [a, a₂], and a2 is not a
al.
a₂}
26. A is a 4 x 3 matrix, A = [a, a2 a3], such that {a,, a2) is
linearly independent and a3 is not in Span {a,, a2}.
27. How many pivot columns must a 7 x 5 matrix have if its
columns are linearly independent? Why?
28. How many pivot columns must a 5 x 7 matrix have if its
columns span R5? Why?
29. Construct 3 x 2 matrices A and B such that Ax = 0 has only
the trivial solution and Bx = 0 has a nontrivial solution.
30. a. Fill in the blank in the following statement: "If A is
an m x n matrix, then the columns of A are linearly
independent if and only if A has pivot columns."
b. Explain why the statement in (a) is true.
Exercises 31 and 32 should be solved without performing row
operations. [Hint: Write Ax = 0 as a vector equation.]
31. Given A =
multiple of
2 3 5
32. Given A =
-5 1
-3 -1
1 0
-4
-4
1
4
-7
is the sum of the first two columns. Find a nontrivial solution
of Ax = 0.
16
5
9 -3
observe that the third column
3 , observe that the first column
3
plus twice the second column equals the third column. Find
a nontrivial solution of Ax = 0.
Each statement in Exercises 33-38 is either true (in all cases)
or false (for at least one example). If false, construct a specific
example to show that the statement is not always true. Such an
example is called a counterexample to the statement. If a statement
is true, give a justification. (One specific example cannot explain
why a statement is always true. You will have to do more work
here than in Exercises 21 and 22.)
33. If v₁,..., V4 are in R4 and v3 = 2v₁ + V2, then {V₁, V2, V3, V4)
is linearly dependent.
34. If V₁,..., V4 are in R4 and v3 = 0, then {V₁, V2, V3, V4) is
linearly dependent.
35. If v₁ and v₂ are in R4 and v₂ is not a scalar multiple of V₁.
then {V₁, V₂} is linearly independent.
36. If V₁,..., V4 are in R4 and v3 is not a linear combination of
V₁, V2, V4, then {V₁, V2, V3, V4} is linearly independent.
37. If V₁...., V4 are in R4 and {V₁, V2, V3} is linearly dependent.
then {V1, V2, V3, V4) is also linearly dependent.
38. If V₁,..., V4 are linearly independent vectors in R4, then
{V1, V2, V3} is also linearly independent. [Hint: Think about
XIVI + X₂V₂ + X3V3 +0 V4 = 0.]
1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5587edcc-e35c-40e1-a9da-86738624fe94%2F23688b7d-0bd5-483e-9479-e4d3ea0af279%2F9sles0n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:62 CHAPTER 1 Linear Equations in Linear Algebra
9. V₁ =
10. V₁ =
11.
13.
-[-]--
-3
2
15.
In Exercises 11-14, find the value(s) of h for which the vectors
are linearly dependent. Justify each answer.
17.
-----
=
10
-3
-
-
-5
-
3
5
3
-5
7
-8
19. 12
-4
-5
BED AND
5
-9
7
6
8
5
h
1.[8]
0
9
6
-6
5
4
3
9
Determine by inspection whether the vectors in Exercises 15-20
are linearly independent. Justify each answer.
[{][B][B][-]
1
12.
14.
16.
2
-9
h
2
HO
-4
7
-3
18.
6
[][]
-2
9
2
20.
4 0₁
In Exercises 21 and 22, mark each statement True or False. Justify
each answer on the basis of a careful reading of the text.
h
[4][3][³][8]
5
-7
-
-2
5
3
0
[8]
21. a. The columns of a matrix A are linearly independent if the
equation Ax = 0 has the trivial solution.
b. If S is a linearly dependent set, then each vector is a linear
combination of the other vectors in S.
c. The columns of any 4 x 5 matrix are linearly dependent.
d. If x and y are linearly independent, and if {x, y, z) is
linearly dependent, then z is in Span {x,y}.
22. /a. Two vectors are linearly dependent if and only if they lie
on a line through the origin.
b. If a set contains fewer vectors than there are entries in the
vectors, then the set is linearly independent.
c. If x and y are linearly independent, and if z is in
Span {x, y), then {x, y, z) is linearly dependent.
d. If a set in R" is linearly dependent, then the set contains
more vectors than there are entries in each vector.
In Exercises 23-26, describe the possible echelon forms of the
matrix. Use the notation of Example 1 in Section 1.2.
23. A is a 3 x 3 matrix with linearly independent columns.
24. A is a 2 x 2 matrix with linearly dependent columns,
25. A is a 4 × 2 matrix, A = [a, a₂], and a2 is not a
al.
a₂}
26. A is a 4 x 3 matrix, A = [a, a2 a3], such that {a,, a2) is
linearly independent and a3 is not in Span {a,, a2}.
27. How many pivot columns must a 7 x 5 matrix have if its
columns are linearly independent? Why?
28. How many pivot columns must a 5 x 7 matrix have if its
columns span R5? Why?
29. Construct 3 x 2 matrices A and B such that Ax = 0 has only
the trivial solution and Bx = 0 has a nontrivial solution.
30. a. Fill in the blank in the following statement: "If A is
an m x n matrix, then the columns of A are linearly
independent if and only if A has pivot columns."
b. Explain why the statement in (a) is true.
Exercises 31 and 32 should be solved without performing row
operations. [Hint: Write Ax = 0 as a vector equation.]
31. Given A =
multiple of
2 3 5
32. Given A =
-5 1
-3 -1
1 0
-4
-4
1
4
-7
is the sum of the first two columns. Find a nontrivial solution
of Ax = 0.
16
5
9 -3
observe that the third column
3 , observe that the first column
3
plus twice the second column equals the third column. Find
a nontrivial solution of Ax = 0.
Each statement in Exercises 33-38 is either true (in all cases)
or false (for at least one example). If false, construct a specific
example to show that the statement is not always true. Such an
example is called a counterexample to the statement. If a statement
is true, give a justification. (One specific example cannot explain
why a statement is always true. You will have to do more work
here than in Exercises 21 and 22.)
33. If v₁,..., V4 are in R4 and v3 = 2v₁ + V2, then {V₁, V2, V3, V4)
is linearly dependent.
34. If V₁,..., V4 are in R4 and v3 = 0, then {V₁, V2, V3, V4) is
linearly dependent.
35. If v₁ and v₂ are in R4 and v₂ is not a scalar multiple of V₁.
then {V₁, V₂} is linearly independent.
36. If V₁,..., V4 are in R4 and v3 is not a linear combination of
V₁, V2, V4, then {V₁, V2, V3, V4} is linearly independent.
37. If V₁...., V4 are in R4 and {V₁, V2, V3} is linearly dependent.
then {V1, V2, V3, V4) is also linearly dependent.
38. If V₁,..., V4 are linearly independent vectors in R4, then
{V1, V2, V3} is also linearly independent. [Hint: Think about
XIVI + X₂V₂ + X3V3 +0 V4 = 0.]
1.
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