1. T: R² R4, 7(e₁) = (3, 1, 3, 1) and T (e₂) = (-5,2,0,0), where e (1, 0) and e₂ = (0, 1). =
1. T: R² R4, 7(e₁) = (3, 1, 3, 1) and T (e₂) = (-5,2,0,0), where e (1, 0) and e₂ = (0, 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
1
![1.9 EXERCISES
In Exercises 1-10, assume that T is a linear transformation. Find
the standard matrix of T.
110
1. T: R² → R¹,T(e₁) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),
where e₁ = (1,0) and e₂ = (0, 1).
TR³ R², T(e₁) = (1,3),
T(e₁) = (1,3), T(e₂) =(4,-7), and
T(e3) = (-5,4), where e₁, e₂2, e3 are the columns of the
3 x 3 identity matrix.
Jugdaya
2.7:
2/
3. T: R² → R² rotates points (about the origin) through 37/2
radians (counterclockwise).qqam od
wodz 200
EU
4. T: R² R² rotates points (about the origin) through -/4
radians (clockwise). [Hint: T (e₁) = (1/√√2, -1/√2).]
17
5. T: R² R2 is a vertical shear transformation that maps e₁
into e₁ - 2e₂ but leaves the vector e2 unchanged.
6. T: R² R2 is a horizontal shear transformation that leaves
e, 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).]
3. T: R2 R2 first reflects points through the horizontal x₁-
axis and then reflects points through the line x2 = X₁.
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.
T: R² R² first reflects points through the vertical x2-axis
In Ex
assur
15.
tr
16.
In E
findi
are r
17.
18.
19.
20.
21.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb08f5b58-11d5-48ec-9a64-2ab5300ee2b0%2Fe4704720-0eb6-4d5b-9d6e-faf80e81f86a%2Fqpkn8dq_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.
110
1. T: R² → R¹,T(e₁) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),
where e₁ = (1,0) and e₂ = (0, 1).
TR³ R², T(e₁) = (1,3),
T(e₁) = (1,3), T(e₂) =(4,-7), and
T(e3) = (-5,4), where e₁, e₂2, e3 are the columns of the
3 x 3 identity matrix.
Jugdaya
2.7:
2/
3. T: R² → R² rotates points (about the origin) through 37/2
radians (counterclockwise).qqam od
wodz 200
EU
4. T: R² R² rotates points (about the origin) through -/4
radians (clockwise). [Hint: T (e₁) = (1/√√2, -1/√2).]
17
5. T: R² R2 is a vertical shear transformation that maps e₁
into e₁ - 2e₂ but leaves the vector e2 unchanged.
6. T: R² R2 is a horizontal shear transformation that leaves
e, 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).]
3. T: R2 R2 first reflects points through the horizontal x₁-
axis and then reflects points through the line x2 = X₁.
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.
T: R² R² first reflects points through the vertical x2-axis
In Ex
assur
15.
tr
16.
In E
findi
are r
17.
18.
19.
20.
21.
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