In Exercises 1-10, assume that T is a linear transformation. Find the standard matrix of T. 1. T: R² → R4, 7(e) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0), where e₁ = (1, 0) and e₂ = (0, 1). 2. T: R³ R², T(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. → 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 into e₁2e₂ 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 -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 e₁ unchanged) and then re- → transformation T In Exercises 15 and 1 assuming that the equa ? X ? ? ? ? 15. 22 16. ? ? ? ? ? 222 ? ? ? X1 X2 In Exercises 17-20, sh finding a matrix that imp are not vectors but are e 17. T(X1, X2, X3, X4) = 18. T(x1, x₂) = (2x₂- 19. T(X1, X2, X3) = (x₁ = 20. T(X1, X2, X3, X4) =
In Exercises 1-10, assume that T is a linear transformation. Find the standard matrix of T. 1. T: R² → R4, 7(e) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0), where e₁ = (1, 0) and e₂ = (0, 1). 2. T: R³ R², T(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. → 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 into e₁2e₂ 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 -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 e₁ unchanged) and then re- → transformation T In Exercises 15 and 1 assuming that the equa ? X ? ? ? ? 15. 22 16. ? ? ? ? ? 222 ? ? ? X1 X2 In Exercises 17-20, sh finding a matrix that imp are not vectors but are e 17. T(X1, X2, X3, X4) = 18. T(x1, x₂) = (2x₂- 19. T(X1, X2, X3) = (x₁ = 20. T(X1, X2, X3, X4) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Ans 7
![In Exercises 1-10, assume that 7 is a linear transformation. Find
the standard matrix of T.
1. T: R² → R4, 7(e) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),
where e₁ = (1, 0) and e₂ = (0, 1).
2. T: R³ → R2², 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.
→
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 e₁- 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 -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 e₁ unchanged) and then re-
transformation T
In Exercises 15 and 10
assuming that the equa
16.
? I ?
?
?
15. ?
?
?
?
?
?
?
?
? ?
X1
X2
In Exercises 17-20, sh
finding a matrix that imp
are not vectors but are e
17. T(X1, X2, X3, X4) =
18. T(x₁, x₂) = (2x₂-
19. T(X1, X2, X3) = (XL
20. T(X1, X2, X3, X4) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6147cd51-fae9-4b86-867f-e7a7fe5750a1%2F8e2c7b07-7bcf-4169-8a63-8b67b0c1ec4e%2Fso27ru_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In Exercises 1-10, assume that 7 is a linear transformation. Find
the standard matrix of T.
1. T: R² → R4, 7(e) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0),
where e₁ = (1, 0) and e₂ = (0, 1).
2. T: R³ → R2², 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.
→
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 e₁- 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 -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 e₁ unchanged) and then re-
transformation T
In Exercises 15 and 10
assuming that the equa
16.
? I ?
?
?
15. ?
?
?
?
?
?
?
?
? ?
X1
X2
In Exercises 17-20, sh
finding a matrix that imp
are not vectors but are e
17. T(X1, X2, X3, X4) =
18. T(x₁, x₂) = (2x₂-
19. T(X1, X2, X3) = (XL
20. T(X1, X2, X3, X4) =
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