5. T: R² R2 is a vertical shear transformation that maps e into e, - 2e₂ but leaves the vector e₂ unchanged.
5. T: R² R2 is a vertical shear transformation that maps e into e, - 2e₂ but leaves the vector e₂ unchanged.
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
Related questions
Question
5
![where e₁ =
= (1,0) and e₂ = (0, 1).
TR¹ R², T(e₁) = (1,3), T(e₂) =(4,-7), and
T(es) = (-5,4), where e₁, ez, e, are the columns of the
3 x 3 identity matrix.
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(e) = (1/√2, -1/√2).]
5. T: R² → R² is a vertical shear transformation that maps e₁
into e, - 2e₂ but leaves the vector e₂ unchanged.
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 -3π/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.
10. T: R² R² first reflects points through the vertical x2-axis
and then rotates points л/2 radians.
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 T(e₁)
and T(e₂) are the vectors shown in the figure. Using the
figure, sketch the vector T(2, 1).
In Exercises 15
assuming that th
15.
16.
?
?
? ?
?
?
?
?
?
?
?
?
In Exercises 17
finding a matrix
are not vectors b
17. T(X₁, X2, X
"(x₁.
18. T(x₁,
x₂) =
19. T(x₁,x2, x
20. T(x₁,x2, X
21. Let T: R
T(x₁, x₂) =
(3,8).
22.2
22. Let T: R
T(x₁, x₂) =
that 7(x) =
In Exercises 23 a
each answer.
23. a. A linear
3.
termine
matrix.
b. If T: R
an angle
c. When t
another.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5495aa8f-e580-46c5-af54-23236303b730%2Fd6f8f1a4-9754-421b-a48b-239fda9a10b5%2Fm9zlknkr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:where e₁ =
= (1,0) and e₂ = (0, 1).
TR¹ R², T(e₁) = (1,3), T(e₂) =(4,-7), and
T(es) = (-5,4), where e₁, ez, e, are the columns of the
3 x 3 identity matrix.
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(e) = (1/√2, -1/√2).]
5. T: R² → R² is a vertical shear transformation that maps e₁
into e, - 2e₂ but leaves the vector e₂ unchanged.
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 -3π/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.
10. T: R² R² first reflects points through the vertical x2-axis
and then rotates points л/2 radians.
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 T(e₁)
and T(e₂) are the vectors shown in the figure. Using the
figure, sketch the vector T(2, 1).
In Exercises 15
assuming that th
15.
16.
?
?
? ?
?
?
?
?
?
?
?
?
In Exercises 17
finding a matrix
are not vectors b
17. T(X₁, X2, X
"(x₁.
18. T(x₁,
x₂) =
19. T(x₁,x2, x
20. T(x₁,x2, X
21. Let T: R
T(x₁, x₂) =
(3,8).
22.2
22. Let T: R
T(x₁, x₂) =
that 7(x) =
In Exercises 23 a
each answer.
23. a. A linear
3.
termine
matrix.
b. If T: R
an angle
c. When t
another.
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