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).
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).
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
![7. T: R²R first rotates points through -3π/4 radian
(clockwise) and then reflects points through the horizontal
x-axis. [Hint: T (e) = (-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 e 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).
T(e₁)
T(e₂)
X₁
14. Let T: R²
→>>>
R2 be a linear transformation with standard
matrix A = [a₁
a2], where a, and a2 are
figure. Using the figure, draw the image of
shown in the
under the
3
finding a
are not ve
17. TX₁
18. T(x1
19. T(x₁
20. T(X₁
21. Let
T(x₁
(3,8)
22. Let
13.
T(x₁
that 7
In Exercis
each answ
a. A
te
m
b. If
an
c. W
an
tra
d. A
R'
e. If
ca
24. a. No
tra
b. Th
ma
the](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F973d6112-791f-4249-91d8-33d0c69fc61e%2F9956689a-0efa-49d8-83a5-7f79014cb78e%2Fx6wlb7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7. T: R²R first rotates points through -3π/4 radian
(clockwise) and then reflects points through the horizontal
x-axis. [Hint: T (e) = (-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 e 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).
T(e₁)
T(e₂)
X₁
14. Let T: R²
→>>>
R2 be a linear transformation with standard
matrix A = [a₁
a2], where a, and a2 are
figure. Using the figure, draw the image of
shown in the
under the
3
finding a
are not ve
17. TX₁
18. T(x1
19. T(x₁
20. T(X₁
21. Let
T(x₁
(3,8)
22. Let
13.
T(x₁
that 7
In Exercis
each answ
a. A
te
m
b. If
an
c. W
an
tra
d. A
R'
e. If
ca
24. a. No
tra
b. Th
ma
the
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