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.
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.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
Related questions
Question
29
![17.
19.
5
0
1·[8]·L
-8
12
-3
[3]
5
4
18.
20.
-2
4
5
CO
3
0
0
In Exercises 21 and 22, mark each statement True or False. Justify
each answer on the basis of a careful reading of the text.
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
22.
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.
32
E
0
e
e
i](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5a5d7bf-20ae-45d2-981f-6a6741ca8eb9%2F3afd7cc0-7ba2-49e5-aa8a-1ab1bc067848%2Fk67hg4m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:17.
19.
5
0
1·[8]·L
-8
12
-3
[3]
5
4
18.
20.
-2
4
5
CO
3
0
0
In Exercises 21 and 22, mark each statement True or False. Justify
each answer on the basis of a careful reading of the text.
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
22.
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.
32
E
0
e
e
i
![17.
19.
5
0
1·[8]·L
-8
12
-3
[3]
5
4
18.
20.
-2
4
5
CO
3
0
0
In Exercises 21 and 22, mark each statement True or False. Justify
each answer on the basis of a careful reading of the text.
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
22.
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.
32
E
0
e
e
i](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5a5d7bf-20ae-45d2-981f-6a6741ca8eb9%2F3afd7cc0-7ba2-49e5-aa8a-1ab1bc067848%2Fbckxsgo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:17.
19.
5
0
1·[8]·L
-8
12
-3
[3]
5
4
18.
20.
-2
4
5
CO
3
0
0
In Exercises 21 and 22, mark each statement True or False. Justify
each answer on the basis of a careful reading of the text.
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
22.
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.
32
E
0
e
e
i
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