5. T: R² R2 is a vertical shear transformation that maps e₁ into e₁2e₂ but leaves the vector e2 unchanged. -
5. T: R² R2 is a vertical shear transformation that maps e₁ into e₁2e₂ but leaves the vector e2 unchanged. -
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
5
![1.9 EXERCISES
In Exercises 1-10, assume that T is a linear transformation. Find
the standard matrix of T.
1. T : R² → R4, 7(e₁) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),
where e₁ = (1, 0) and e₂ = (0, 1).
:
->
2.7: R³ R², T(e₁) = (1,3), 7(e₂) =(4,-7), and
T(e3)= (-5,4), where e₁, e2, e3 are the columns of the
3 x 3 identity matrix.
3. T: R2 R2 rotates points (about the origin) through 37/2
uradians (counterclockwise). de m
→>>
4. T R2 R2 rotates points (about the origin) through -л/4
radians (clockwise). [Hint: T(e₁) = (1/√2, -1/√2).]
www.
NRICS
5. T: R² R2 is a vertical shear transformation that maps e₁
into e₁ - 2e₂ but leaves the vector e₂ unchanged.
ody no render wall on
6. T: R²
R2 is a horizontal shear transformation that leaves
e₁ unchanged and maps e₂ into e₂ + 3e₁.
→>>
7. T: R² R² first rotates points through -3/4 radian
(clockwise) and then reflects points through the horizontal
x₁-axis. [Hint: T(e₁) = (-1/√2, 1/√2).]
8. T: R² → R² first reflects points through the horizontal x₁-
axis and then reflects points through the line x2 = X1.
9. T: R² R² first performs a horizontal shear that trans-
forms e2 into e2 - 2e₁ (leaving e, unchanged) and then re-
flects points through the line x₂ = -X₁.
0. T: R² R2 first reflects points through the vertical x2-axis
and then rotates points л/2 radians.
1. A linear transformation T: R2 R2 first reflects points
through the x₁-axis and then reflects points through the x2-
axis. Show that T can also be described as a linear transfor-
mation that rotates points about the origin. What is the angle
of that rotation?
2. Show that the transformation in Exercise 8 is merely a rota-
tion about the origin. What is the angle of the rotation?
3. Let T: R2 R2 be the linear transformation such that T(e₁)
and T(₂) are the vectors shown in the figure. Using the
figure, sketch the vector T(2, 1).
T(e,)
x2
T(e₂)
X1
4. Let T: R² → R² be a linear transformation with standard
matrix A = [a₁ a2], where a, and a2 are shown in the
under the
figure. Using the figure, draw the image of [3]
transt
transformation T.
anol d
16.
? ?
?
?
15. ?
?
T
In Exercises 15 and 16, fill in the missing entries of the matrix,
assuming that the equation holds for all values of the variables.
3x₁ - 2x3
4x1
x1 - x₂ + x3
th?
X1
X2
I-
?
x2
?
? ?
?
ift
?
X1
13]-[
=
?
x2.
X3
a2
x1 - x2
-2x1 + x₂
X1
X1
a₁
In Exercises 17-20, show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x₁, x2,...
are not vectors but are entries in vectors.
17. T(X1, X2, X3, X4) = (0, x₁ + x2, x2 + x3, x3 + x4)
18./T(x₁, x2) = (2x2 - 3x1, x₁ - 4x2, 0, x₂)
19. T(X₁, X2, X3) = (x1 - 5x2 + 4x3, x2 - 6x3)
20. T(X1, X2, X3, X4) = 2x1 + 3x3 - 4x4 (T: R4 → R)
21. Let T: R2 R2 be a linear transformation such that
T(x₁, x₂) = (x₁ + x2, 4x₁ + 5x2). Find x such that T(x) =
(3,8).
22. Let T R² R³ be a linear transformation such that
T(x₁, x2) = (x₁ - 2x2, -x1 + 3x2, 3x1 - 2x₂). Find x such
that T(x) = (-1,4,9).
0030
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
m
237.
23. a. A linear transformation T: R" → R" is completely de-
termined by its effect on the columns of the n x n identity
matrix.
b. If T: R² R2 rotates vectors about the origin through
an angle , then T is a linear transformation.
c. When two linear transformations are performed one after
another, the combined effect may not always be a linear
transformation.
d. A mapping T: R" → R" is onto Rm if every vector x in
R"
maps onto some vector in Rm.
e. If A is a 3 x 2 matrix, then the transformation X→→ Ax
cannot be one-to-one.
24. a. Not every linear transformation from R" to R" is a matrix
transformation.
m
b. The columns of the standard matrix for a linear transfor-
mation from R" to R" are the images of the columns of
the nxn identity matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a737fc6-15a5-4a44-9fac-a744876436b7%2Ff24ab0be-0306-44c9-88a6-77bca94c2390%2F81hl0dv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1.9 EXERCISES
In Exercises 1-10, assume that T is a linear transformation. Find
the standard matrix of T.
1. T : R² → R4, 7(e₁) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),
where e₁ = (1, 0) and e₂ = (0, 1).
:
->
2.7: R³ R², T(e₁) = (1,3), 7(e₂) =(4,-7), and
T(e3)= (-5,4), where e₁, e2, e3 are the columns of the
3 x 3 identity matrix.
3. T: R2 R2 rotates points (about the origin) through 37/2
uradians (counterclockwise). de m
→>>
4. T R2 R2 rotates points (about the origin) through -л/4
radians (clockwise). [Hint: T(e₁) = (1/√2, -1/√2).]
www.
NRICS
5. T: R² R2 is a vertical shear transformation that maps e₁
into e₁ - 2e₂ but leaves the vector e₂ unchanged.
ody no render wall on
6. T: R²
R2 is a horizontal shear transformation that leaves
e₁ unchanged and maps e₂ into e₂ + 3e₁.
→>>
7. T: R² R² first rotates points through -3/4 radian
(clockwise) and then reflects points through the horizontal
x₁-axis. [Hint: T(e₁) = (-1/√2, 1/√2).]
8. T: R² → R² first reflects points through the horizontal x₁-
axis and then reflects points through the line x2 = X1.
9. T: R² R² first performs a horizontal shear that trans-
forms e2 into e2 - 2e₁ (leaving e, unchanged) and then re-
flects points through the line x₂ = -X₁.
0. T: R² R2 first reflects points through the vertical x2-axis
and then rotates points л/2 radians.
1. A linear transformation T: R2 R2 first reflects points
through the x₁-axis and then reflects points through the x2-
axis. Show that T can also be described as a linear transfor-
mation that rotates points about the origin. What is the angle
of that rotation?
2. Show that the transformation in Exercise 8 is merely a rota-
tion about the origin. What is the angle of the rotation?
3. Let T: R2 R2 be the linear transformation such that T(e₁)
and T(₂) are the vectors shown in the figure. Using the
figure, sketch the vector T(2, 1).
T(e,)
x2
T(e₂)
X1
4. Let T: R² → R² be a linear transformation with standard
matrix A = [a₁ a2], where a, and a2 are shown in the
under the
figure. Using the figure, draw the image of [3]
transt
transformation T.
anol d
16.
? ?
?
?
15. ?
?
T
In Exercises 15 and 16, fill in the missing entries of the matrix,
assuming that the equation holds for all values of the variables.
3x₁ - 2x3
4x1
x1 - x₂ + x3
th?
X1
X2
I-
?
x2
?
? ?
?
ift
?
X1
13]-[
=
?
x2.
X3
a2
x1 - x2
-2x1 + x₂
X1
X1
a₁
In Exercises 17-20, show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x₁, x2,...
are not vectors but are entries in vectors.
17. T(X1, X2, X3, X4) = (0, x₁ + x2, x2 + x3, x3 + x4)
18./T(x₁, x2) = (2x2 - 3x1, x₁ - 4x2, 0, x₂)
19. T(X₁, X2, X3) = (x1 - 5x2 + 4x3, x2 - 6x3)
20. T(X1, X2, X3, X4) = 2x1 + 3x3 - 4x4 (T: R4 → R)
21. Let T: R2 R2 be a linear transformation such that
T(x₁, x₂) = (x₁ + x2, 4x₁ + 5x2). Find x such that T(x) =
(3,8).
22. Let T R² R³ be a linear transformation such that
T(x₁, x2) = (x₁ - 2x2, -x1 + 3x2, 3x1 - 2x₂). Find x such
that T(x) = (-1,4,9).
0030
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
m
237.
23. a. A linear transformation T: R" → R" is completely de-
termined by its effect on the columns of the n x n identity
matrix.
b. If T: R² R2 rotates vectors about the origin through
an angle , then T is a linear transformation.
c. When two linear transformations are performed one after
another, the combined effect may not always be a linear
transformation.
d. A mapping T: R" → R" is onto Rm if every vector x in
R"
maps onto some vector in Rm.
e. If A is a 3 x 2 matrix, then the transformation X→→ Ax
cannot be one-to-one.
24. a. Not every linear transformation from R" to R" is a matrix
transformation.
m
b. The columns of the standard matrix for a linear transfor-
mation from R" to R" are the images of the columns of
the nxn identity matrix.
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