THE 10 (and no row operations) to find scalars C₁, C₂, C3 such that 25. Note that -7 -3 B-080 [] -3 = C₁ 5 + C₂ -2 +C3 5 4 -3 1 5-2 5 2-3 -6 10 -3 Use this fact
THE 10 (and no row operations) to find scalars C₁, C₂, C3 such that 25. Note that -7 -3 B-080 [] -3 = C₁ 5 + C₂ -2 +C3 5 4 -3 1 5-2 5 2-3 -6 10 -3 Use this fact
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
25
![42 CHAPTER 1 Linear Equations in Linear Algebra
f. If A is an m x n matrix whose columns do not span R",
then the equation Ax=b is inconsistent for some b in
Rm.
4 -3 1
5 -2
3-E
= -3
10
2 -3
2
-6
(and no row operations) to find scalars C₁, C2, C3 such that
-7
4
-3
1
-3
5
5
10
25. Note that
26. Let u =
= C1
7
2
5
-6
H
5
2 , V =
+C₂-2
3
G
3
6
3]
It can be shown that 3u-5v-w= 0. Use this fact (and
no row operations) to find x₁ and x2 that satisfy the equation
3
6
X1
1
I
]-[1]
=
X2
3
5 -
2
-3 5 -4 01 97
5
8
1 -2 -4
+ C3
and w =
-3
H
d
27. Let 9₁, 92, 93, and v represent vectors in R5, and let x₁, x2,
and x3 denote scalars. Write the following vector equation as
a matrix equation. Identify any symbols you choose to use.
X191 + x₂9₂ + x393 = V
28. Rewrite the (numerical) matrix equation below in symbolic
form as a vector equation, using symbols V₁, V2, ... for the
vectors and C₁, C2,... for scalars. Define what each symbol
represents, using the data given in the matrix equation.
-3
2
01
4
-1
2
Use this fact
8
=[-2]
29. Construct a 3 x 3 matrix,
not in echelon form, whose
columns span R³. Show that the matrix you construct has the
mi notlizoq
desired property.
Vink
nabora
30. Construct a 3×3 matrix, not in echelon form, whose
ad columns do not span R³. Show that the matrix you construct
has the desired
property.
31. Let A be a 3 x 2 matrix. Explain why the equation Ax = b
cannot be consistent for all b in R³. Generalize your
WOT VIVe si noidizoo
32. Could a set of three vectors in R* span all of R4? Explain.
What about n vectors in R" when n is less than m?
33. Suppose A is a 4 x 3 matrix and b is a vector in R4 with the
property that Ax = b has a unique solution. What can you say
about the reduced echelon form of A? Justify your answer.
34. Suppose A is a 3 x 3 matrix and b is a vector in R³ with the
property that Ax = b has a unique solution. Explain why the
columns of A must span
R³.
35. Let A be a 3 x 4 matrix, let y, and y2 be vectors in R³, and
Ax₁ and y₂ =
let w = y₁ + y₂. Suppose y₁ =
Ax₂ for some
vectors X₁ and x2 in R4. What fact allows you to conclude that
the system Ax = w is consistent? (Note: X₁ and x₂ denote
vectors, not scalar entries in vectors.)
oth ovine n
36. Let A be a 5 x 3 matrix, let y be a vector in R³, and let z
be a vector in R5. Suppose Ay
= z. What fact allows you to
conclude that the system Ax = 4z is consistent?
argument to the case of an arbitrary A with more rows than
columns.
[M] In Exercises 37-40, determine if the columns of the matrix
span R4.
37.
39.
40.
прот
-5 -3
10
9
Fi 12
-9
-6
4
8
-7
11
-3
2 -5
4
8
-9
7
2 15
=
-7
11 -9
4
-8 7
11 -7
3
-6 10 -5
-2
-8
11 -6 -7
5 6
7 -7 -9
4 18
38.
5
-3
-9
12
13
-9
-6
7
5 −7
6
-8
4
-4
-9
011
Abne
$
ттад
-4
-7
anar'
9
5
-9 -9
yd bonnage
be show the
41. [M] Find a column of the matrix in Exercise 39 that can be
deleted and yet have the remaining matrix columns still span
R4.
42. [M] Find a column of the matrix in Exercise 40 that can be
deleted and yet have the remaining matrix columns still span
R4. Can you delete more than one column?
ion ydw 10](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc9596d36-d9f8-4224-a071-c71202663891%2F14fe9651-1338-4352-9ab7-6078f48b0d4f%2Fagf8uik_processed.jpeg&w=3840&q=75)
Transcribed Image Text:42 CHAPTER 1 Linear Equations in Linear Algebra
f. If A is an m x n matrix whose columns do not span R",
then the equation Ax=b is inconsistent for some b in
Rm.
4 -3 1
5 -2
3-E
= -3
10
2 -3
2
-6
(and no row operations) to find scalars C₁, C2, C3 such that
-7
4
-3
1
-3
5
5
10
25. Note that
26. Let u =
= C1
7
2
5
-6
H
5
2 , V =
+C₂-2
3
G
3
6
3]
It can be shown that 3u-5v-w= 0. Use this fact (and
no row operations) to find x₁ and x2 that satisfy the equation
3
6
X1
1
I
]-[1]
=
X2
3
5 -
2
-3 5 -4 01 97
5
8
1 -2 -4
+ C3
and w =
-3
H
d
27. Let 9₁, 92, 93, and v represent vectors in R5, and let x₁, x2,
and x3 denote scalars. Write the following vector equation as
a matrix equation. Identify any symbols you choose to use.
X191 + x₂9₂ + x393 = V
28. Rewrite the (numerical) matrix equation below in symbolic
form as a vector equation, using symbols V₁, V2, ... for the
vectors and C₁, C2,... for scalars. Define what each symbol
represents, using the data given in the matrix equation.
-3
2
01
4
-1
2
Use this fact
8
=[-2]
29. Construct a 3 x 3 matrix,
not in echelon form, whose
columns span R³. Show that the matrix you construct has the
mi notlizoq
desired property.
Vink
nabora
30. Construct a 3×3 matrix, not in echelon form, whose
ad columns do not span R³. Show that the matrix you construct
has the desired
property.
31. Let A be a 3 x 2 matrix. Explain why the equation Ax = b
cannot be consistent for all b in R³. Generalize your
WOT VIVe si noidizoo
32. Could a set of three vectors in R* span all of R4? Explain.
What about n vectors in R" when n is less than m?
33. Suppose A is a 4 x 3 matrix and b is a vector in R4 with the
property that Ax = b has a unique solution. What can you say
about the reduced echelon form of A? Justify your answer.
34. Suppose A is a 3 x 3 matrix and b is a vector in R³ with the
property that Ax = b has a unique solution. Explain why the
columns of A must span
R³.
35. Let A be a 3 x 4 matrix, let y, and y2 be vectors in R³, and
Ax₁ and y₂ =
let w = y₁ + y₂. Suppose y₁ =
Ax₂ for some
vectors X₁ and x2 in R4. What fact allows you to conclude that
the system Ax = w is consistent? (Note: X₁ and x₂ denote
vectors, not scalar entries in vectors.)
oth ovine n
36. Let A be a 5 x 3 matrix, let y be a vector in R³, and let z
be a vector in R5. Suppose Ay
= z. What fact allows you to
conclude that the system Ax = 4z is consistent?
argument to the case of an arbitrary A with more rows than
columns.
[M] In Exercises 37-40, determine if the columns of the matrix
span R4.
37.
39.
40.
прот
-5 -3
10
9
Fi 12
-9
-6
4
8
-7
11
-3
2 -5
4
8
-9
7
2 15
=
-7
11 -9
4
-8 7
11 -7
3
-6 10 -5
-2
-8
11 -6 -7
5 6
7 -7 -9
4 18
38.
5
-3
-9
12
13
-9
-6
7
5 −7
6
-8
4
-4
-9
011
Abne
$
ттад
-4
-7
anar'
9
5
-9 -9
yd bonnage
be show the
41. [M] Find a column of the matrix in Exercise 39 that can be
deleted and yet have the remaining matrix columns still span
R4.
42. [M] Find a column of the matrix in Exercise 40 that can be
deleted and yet have the remaining matrix columns still span
R4. Can you delete more than one column?
ion ydw 10
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