9. T: R² R2 first performs a horizontal shear that trans- forms e2 into e2 - 2e₁ (leaving e₁ unchanged) and then re- flects points through the line x2 = -X₁. .PE
9. T: R² R2 first performs a horizontal shear that trans- forms e2 into e2 - 2e₁ (leaving e₁ unchanged) and then re- flects points through the line x2 = -X₁. .PE
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Question
9
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1.9 EXERCISES
In Exercises 1-10, assume that T is a linear transformation. Find
the standard matrix of T.
1. T: R² → R¹, 7(e) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),
where e₁ = (1, 0) and e₂ = (0, 1).
→
2.7: R³ R², 7(e) = (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.b
3. T: R² R2 rotates points (about the origin) through 37/2
radians (counterclockwise).
4. T: R² → R2 rotates points (about the origin) through -/4
radians (clockwise). [Hint: T(e₁) = (1/√2, -1/√2).]
5. T: R² → R2 is a vertical shear transformation that maps e₁
into e₁ - 2e₂ but leaves the vector e2 unchanged.
Tusi,
noland 163nil od
dotox3
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).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 e₁ unchanged) and then re-
flects points through the line x2 = -X₁. 01
Ch
2
→>>>
10. T: R² R2 first reflects points through the vertical x2-axis
ASTR
and then rotates points л/2 radians.
AIS
11. 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?
12. Show that the transformation in Exercise 8 is merely a rota-
tion about the origin. What is the angle of the rotation?
13. Let T: R2 R2 be the linear transformation such that 7(e₁)
and 7(e₂) are the vectors shown in the figure. Using the
figure, sketch the vector T(2, 1).
od
bas 19-almi 19 esgnedo eixe-cx S
x2
01
T(e₂).
15.
transformation T.
transt
16.
In Exercises 15 and 16, fill in the m
assuming that the equation holds for
?
?
?
?
?
?
?
?
?
?
?
?
X1
18-0
x2.
X3
X2
?
?
2
=
[*] [
X2
22. Let T: R²
22.1
3x
X1 -
X1 - X2
-2x₁ + x
X1
In Exercises 17-20, show that T is
finding a matrix that implements the 1
are not vectors but are entries in vec
17. T(X1, X2, X3, X4) = (0, x₁ + x₂-
18./T(x1, x2) = (2x2-3x₁, x₁-4
19. T(X1, X2, X3) = (x₁ - 5x₂ + 43
20. T(X1, X2, X3, X4) = 2x1 + 3x3 -
21. Let T: R2 R2 be a linea
T(x₁, x₂) =
(3,8).
(x₁ + x₂, 4x₁ + 5
R³ be a linea
T(x₁, x₂) = (x₁ - 2x2,-x₁ +
that T(x) = (-1,4,9).
In Exercises 23 and 24, mark each s
each answer.
23. a. A linear transformation T
termined by its effect on the
matrix.
b. If T: R² → R² rotates ve
an angle o, then T is a line
c. When two linear transform
another, the combined effe
transformation.
d. A mapping T:R" →R"
R" maps onto some vector](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5495aa8f-e580-46c5-af54-23236303b730%2Fd8502d0e-1778-4fcb-870c-5bc5ef2ee86e%2Fsvsfkn_processed.jpeg&w=3840&q=75)
Transcribed Image Text:P
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1
1.9 EXERCISES
In Exercises 1-10, assume that T is a linear transformation. Find
the standard matrix of T.
1. T: R² → R¹, 7(e) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),
where e₁ = (1, 0) and e₂ = (0, 1).
→
2.7: R³ R², 7(e) = (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.b
3. T: R² R2 rotates points (about the origin) through 37/2
radians (counterclockwise).
4. T: R² → R2 rotates points (about the origin) through -/4
radians (clockwise). [Hint: T(e₁) = (1/√2, -1/√2).]
5. T: R² → R2 is a vertical shear transformation that maps e₁
into e₁ - 2e₂ but leaves the vector e2 unchanged.
Tusi,
noland 163nil od
dotox3
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).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 e₁ unchanged) and then re-
flects points through the line x2 = -X₁. 01
Ch
2
→>>>
10. T: R² R2 first reflects points through the vertical x2-axis
ASTR
and then rotates points л/2 radians.
AIS
11. 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?
12. Show that the transformation in Exercise 8 is merely a rota-
tion about the origin. What is the angle of the rotation?
13. Let T: R2 R2 be the linear transformation such that 7(e₁)
and 7(e₂) are the vectors shown in the figure. Using the
figure, sketch the vector T(2, 1).
od
bas 19-almi 19 esgnedo eixe-cx S
x2
01
T(e₂).
15.
transformation T.
transt
16.
In Exercises 15 and 16, fill in the m
assuming that the equation holds for
?
?
?
?
?
?
?
?
?
?
?
?
X1
18-0
x2.
X3
X2
?
?
2
=
[*] [
X2
22. Let T: R²
22.1
3x
X1 -
X1 - X2
-2x₁ + x
X1
In Exercises 17-20, show that T is
finding a matrix that implements the 1
are not vectors but are entries in vec
17. T(X1, X2, X3, X4) = (0, x₁ + x₂-
18./T(x1, x2) = (2x2-3x₁, x₁-4
19. T(X1, X2, X3) = (x₁ - 5x₂ + 43
20. T(X1, X2, X3, X4) = 2x1 + 3x3 -
21. Let T: R2 R2 be a linea
T(x₁, x₂) =
(3,8).
(x₁ + x₂, 4x₁ + 5
R³ be a linea
T(x₁, x₂) = (x₁ - 2x2,-x₁ +
that T(x) = (-1,4,9).
In Exercises 23 and 24, mark each s
each answer.
23. a. A linear transformation T
termined by its effect on the
matrix.
b. If T: R² → R² rotates ve
an angle o, then T is a line
c. When two linear transform
another, the combined effe
transformation.
d. A mapping T:R" →R"
R" maps onto some vector
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