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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
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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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.
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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