13. Let u = A = [4] and 4-[-28] 1 1 6 Is u in the plane R³ spanned by the columns of A? (See the figure.) Why or why not? 01 u? EL 11 -u? * Where is u? ninima od over orrespondok o form JA Plane spanned by the columns of A IS UTAM bns batolab
13. Let u = A = [4] and 4-[-28] 1 1 6 Is u in the plane R³ spanned by the columns of A? (See the figure.) Why or why not? 01 u? EL 11 -u? * Where is u? ninima od over orrespondok o form JA Plane spanned by the columns of A IS UTAM bns batolab
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
13
![ens
ly
3
as
in
ed
and
(a)
ted
Do
(d)
en
dj
nit,
7. XI
11. A =
4
12./A =
-4
13. Let u =
9. 3x₁ + x₂ - 5x3 = 9
Xx₂ + 4x3 = 0
+ x2
Cle
4
ZI
8. ²[-2] + ² [ 3 ] + ² [¯ ] + ²[8]-[8]
4
14. Let u =
o tr
In Exercises 9 and 10, write the system first as a vector equation
and then as a matrix equation.
0
−2
15. Let A =
Given A and b in Exercises 11 and 12, write the augmented matrix
for the linear system that corresponds to the matrix equation
Ax = b. Then solve the system and write the solution as a vector.
1
-3
0
-5
3
-5
-
124
1
−4
0
4
4
2
-1
5
Where is u?
2
-3
-[-]-
2
5,b=
−3
+ x3
1
2
3
and A =
u?
1
b= =
7
8
0
2
16. Repeat Exercise 15: A =
10. 8x₁ - x₂ = 4
5x₁ + 4x₂ = 1
X₁ - 3x₂ = 2
spanned by the columns of A? (See the figure.) Why or why
not?
OL
u?
5
and A = 0
6
-8
=
-4
0
-2
m2
9
3 -5
-2 6
1 1
0
121 911
di
2 -1
= [ ²3 ] and b = [ 1 2 ]
-6
b₂
-3
F
Is u in the plane R
8
1
130
Huloo
doo sno ni sio 15 nov na RO
of R³ spanned by the columns of A? Why or why not?
Plane spanned by
the columns of A
DRM IA
EURO
og bns befolab
7
-1. Is u in the subset
boistol
Ax = b does not have a solution for all possible b, and
describe the set of all b for which Ax = b does have a
solution.
Show that the equation
1 -3 -4
2 6,b=
b₁
b2
--]
=
b3
5 -1 -8
Exercises 17-20 refer to the matrices A and B below. Make
appropriate calculations that justify your answers and mention an
appropriate theorem.
A =
1
-1
0
2
an
3
0
0 3
-1 -1 1
-4
0
1.4 The Matrix Equation Ax = b 41
2-8
3 -1
17. How many rows of A contain a pivot position? Does the
equation Ax=b have a solution for each b in R4?
21. Let v₁ =
18. Do the columns of B span R4? Does the equation Bx = y
have a solution for each y in R4?
19. Can each vector in R4 be written as a linear combination of
the columns of the matrix A above? Do the columns of A
span R4?
20. Can every vector in R4 be written as a linear combination of
be the columns of the matrix B above? Do the columns of B
span R³?land z brs bouto (osqu Wolon
00
1
0
-1
0
ao
Joue.
TOY
B =
224 Let V₁ = -2
, V₂ =
1
0
1
0, V₂ =
3 -2
2
1
1 -5
2 -3 7
-2-8 2 -1
외
0
1
Does (V1, V2, V3} span R4? Why or why not?
of seconds to cloduiya yms yuiesbl nohup
0
- [9].
Does {V1, V2, V3} span R³? Why or why not?
0
-3
8
, V3 =
Arb
, V3 =
0
0
=
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
4
-1
-5
23. a. The equation Ax = b is referred to as a vector equation.
b. A vector b is a linear combination of the columns of a
matrix A if and only if the equation Ax = b has at least
one solution.
not nols
X16M
c. The equation Ax = b is consistent if the augmented ma-
trix [ A b] has a pivot position in every row.
Com
d.
The first entry in the product Ax is a sum of products.
If the columns of an m x n matrix A span R", then the
equation Ax = b is consistent for each b in Rm.
f. If A is an m x n matrix and if the equation Ax = b is
inconsistent for some b in Rm, then A cannot have a pivo
position in every row.
24./a. Every matrix equation Ax = b corresponds to a vecto
equation with the same solution set.
b. Any linear combination of vectors can always be writte
in the form Ax for a suitable matrix A and vector x.
c. The solution set of a linear system whose augment
matrix is [a₁ a2
a2 a3
a3 b] is the same as the soluti
set of Ax=b, if A = [a₁ a2 a3 ].
d. If the equation Ax = b is inconsistent, then b is not in
set spanned by the columns of A.
e. If the augmented matrix [A b] has a pivot position
every row, then the equation Ax = b is inconsistent.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9596d36-d9f8-4224-a071-c71202663891%2Fdeb26cc8-bf3b-4a8f-8fe8-801dcb133857%2Ffwkfsos_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ens
ly
3
as
in
ed
and
(a)
ted
Do
(d)
en
dj
nit,
7. XI
11. A =
4
12./A =
-4
13. Let u =
9. 3x₁ + x₂ - 5x3 = 9
Xx₂ + 4x3 = 0
+ x2
Cle
4
ZI
8. ²[-2] + ² [ 3 ] + ² [¯ ] + ²[8]-[8]
4
14. Let u =
o tr
In Exercises 9 and 10, write the system first as a vector equation
and then as a matrix equation.
0
−2
15. Let A =
Given A and b in Exercises 11 and 12, write the augmented matrix
for the linear system that corresponds to the matrix equation
Ax = b. Then solve the system and write the solution as a vector.
1
-3
0
-5
3
-5
-
124
1
−4
0
4
4
2
-1
5
Where is u?
2
-3
-[-]-
2
5,b=
−3
+ x3
1
2
3
and A =
u?
1
b= =
7
8
0
2
16. Repeat Exercise 15: A =
10. 8x₁ - x₂ = 4
5x₁ + 4x₂ = 1
X₁ - 3x₂ = 2
spanned by the columns of A? (See the figure.) Why or why
not?
OL
u?
5
and A = 0
6
-8
=
-4
0
-2
m2
9
3 -5
-2 6
1 1
0
121 911
di
2 -1
= [ ²3 ] and b = [ 1 2 ]
-6
b₂
-3
F
Is u in the plane R
8
1
130
Huloo
doo sno ni sio 15 nov na RO
of R³ spanned by the columns of A? Why or why not?
Plane spanned by
the columns of A
DRM IA
EURO
og bns befolab
7
-1. Is u in the subset
boistol
Ax = b does not have a solution for all possible b, and
describe the set of all b for which Ax = b does have a
solution.
Show that the equation
1 -3 -4
2 6,b=
b₁
b2
--]
=
b3
5 -1 -8
Exercises 17-20 refer to the matrices A and B below. Make
appropriate calculations that justify your answers and mention an
appropriate theorem.
A =
1
-1
0
2
an
3
0
0 3
-1 -1 1
-4
0
1.4 The Matrix Equation Ax = b 41
2-8
3 -1
17. How many rows of A contain a pivot position? Does the
equation Ax=b have a solution for each b in R4?
21. Let v₁ =
18. Do the columns of B span R4? Does the equation Bx = y
have a solution for each y in R4?
19. Can each vector in R4 be written as a linear combination of
the columns of the matrix A above? Do the columns of A
span R4?
20. Can every vector in R4 be written as a linear combination of
be the columns of the matrix B above? Do the columns of B
span R³?land z brs bouto (osqu Wolon
00
1
0
-1
0
ao
Joue.
TOY
B =
224 Let V₁ = -2
, V₂ =
1
0
1
0, V₂ =
3 -2
2
1
1 -5
2 -3 7
-2-8 2 -1
외
0
1
Does (V1, V2, V3} span R4? Why or why not?
of seconds to cloduiya yms yuiesbl nohup
0
- [9].
Does {V1, V2, V3} span R³? Why or why not?
0
-3
8
, V3 =
Arb
, V3 =
0
0
=
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
4
-1
-5
23. a. The equation Ax = b is referred to as a vector equation.
b. A vector b is a linear combination of the columns of a
matrix A if and only if the equation Ax = b has at least
one solution.
not nols
X16M
c. The equation Ax = b is consistent if the augmented ma-
trix [ A b] has a pivot position in every row.
Com
d.
The first entry in the product Ax is a sum of products.
If the columns of an m x n matrix A span R", then the
equation Ax = b is consistent for each b in Rm.
f. If A is an m x n matrix and if the equation Ax = b is
inconsistent for some b in Rm, then A cannot have a pivo
position in every row.
24./a. Every matrix equation Ax = b corresponds to a vecto
equation with the same solution set.
b. Any linear combination of vectors can always be writte
in the form Ax for a suitable matrix A and vector x.
c. The solution set of a linear system whose augment
matrix is [a₁ a2
a2 a3
a3 b] is the same as the soluti
set of Ax=b, if A = [a₁ a2 a3 ].
d. If the equation Ax = b is inconsistent, then b is not in
set spanned by the columns of A.
e. If the augmented matrix [A b] has a pivot position
every row, then the equation Ax = b is inconsistent.
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