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9. T: R² R2 first performs a horizontal shear that trans- forms e2 into e2 - 2e₁ (leaving e₁ unchanged) and then re- flects points through the line x2 = -X₁. .PE

9. T: R² R2 first performs a horizontal shear that trans- forms e2 into e2 - 2e₁ (leaving e₁ unchanged) and then re- flects points through the line x2 = -X₁. .PE

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
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P ach 0,6 read R Se en TIE IS. t $5 ea 1 1.9 EXERCISES In Exercises 1-10, assume that T is a linear transformation. Find the standard matrix of T. 1. T: R² → R¹, 7(e) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0), where e₁ = (1, 0) and e₂ = (0, 1). → 2.7: R³ R², 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.b 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 e₁ into e₁ - 2e₂ but leaves the vector e2 unchanged. Tusi, noland 163nil od dotox3 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).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- flects points through the line x2 = -X₁. 01 Ch 2 →>>> 10. T: R² R2 first reflects points through the vertical x2-axis ASTR and then rotates points л/2 radians. AIS 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 7(e₁) and 7(e₂) are the vectors shown in the figure. Using the figure, sketch the vector T(2, 1). od bas 19-almi 19 esgnedo eixe-cx S x2 01 T(e₂). 15. transformation T. transt 16. In Exercises 15 and 16, fill in the m assuming that the equation holds for ? ? ? ? ? ? ? ? ? ? ? ? X1 18-0 x2. X3 X2 ? ? 2 = [*] [ X2 22. Let T: R² 22.1 3x X1 - X1 - X2 -2x₁ + x X1 In Exercises 17-20, show that T is finding a matrix that implements the 1 are not vectors but are entries in vec 17. T(X1, X2, X3, X4) = (0, x₁ + x₂- 18./T(x1, x2) = (2x2-3x₁, x₁-4 19. T(X1, X2, X3) = (x₁ - 5x₂ + 43 20. T(X1, X2, X3, X4) = 2x1 + 3x3 - 21. Let T: R2 R2 be a linea T(x₁, x₂) = (3,8). (x₁ + x₂, 4x₁ + 5 R³ be a linea T(x₁, x₂) = (x₁ - 2x2,-x₁ + that T(x) = (-1,4,9). In Exercises 23 and 24, mark each s each answer. 23. a. A linear transformation T termined by its effect on the matrix. b. If T: R² → R² rotates ve an angle o, then T is a line c. When two linear transform another, the combined effe transformation. d. A mapping T:R" →R" R" maps onto some vector
Transcribed Image Text:P ach 0,6 read R Se en TIE IS. t $5 ea 1 1.9 EXERCISES In Exercises 1-10, assume that T is a linear transformation. Find the standard matrix of T. 1. T: R² → R¹, 7(e) = (3, 1, 3, 1) and 7(e₂) = (-5,2,0,0), where e₁ = (1, 0) and e₂ = (0, 1). → 2.7: R³ R², 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.b 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 e₁ into e₁ - 2e₂ but leaves the vector e2 unchanged. Tusi, noland 163nil od dotox3 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).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- flects points through the line x2 = -X₁. 01 Ch 2 →>>> 10. T: R² R2 first reflects points through the vertical x2-axis ASTR and then rotates points л/2 radians. AIS 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 7(e₁) and 7(e₂) are the vectors shown in the figure. Using the figure, sketch the vector T(2, 1). od bas 19-almi 19 esgnedo eixe-cx S x2 01 T(e₂). 15. transformation T. transt 16. In Exercises 15 and 16, fill in the m assuming that the equation holds for ? ? ? ? ? ? ? ? ? ? ? ? X1 18-0 x2. X3 X2 ? ? 2 = [*] [ X2 22. Let T: R² 22.1 3x X1 - X1 - X2 -2x₁ + x X1 In Exercises 17-20, show that T is finding a matrix that implements the 1 are not vectors but are entries in vec 17. T(X1, X2, X3, X4) = (0, x₁ + x₂- 18./T(x1, x2) = (2x2-3x₁, x₁-4 19. T(X1, X2, X3) = (x₁ - 5x₂ + 43 20. T(X1, X2, X3, X4) = 2x1 + 3x3 - 21. Let T: R2 R2 be a linea T(x₁, x₂) = (3,8). (x₁ + x₂, 4x₁ + 5 R³ be a linea T(x₁, x₂) = (x₁ - 2x2,-x₁ + that T(x) = (-1,4,9). In Exercises 23 and 24, mark each s each answer. 23. a. A linear transformation T termined by its effect on the matrix. b. If T: R² → R² rotates ve an angle o, then T is a line c. When two linear transform another, the combined effe transformation. d. A mapping T:R" →R" R" maps onto some vector
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