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T is the reflection in the vector w = (3, 1) in R²: T(v) = 2 proj„v – v, v = (1, 4).

T is the reflection in the vector w = (3, 1) in R²: T(v) = 2 proj„v – v, v = (1, 4).

College Algebra
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter6: Matrices And Determinants
Section: Chapter Questions
Problem 5P
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7.Please solve only Question#22 parts (a) and (b)
Finding the Standard Matrix and the Image In Exercises 11-22, (a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image. 11. Tis the reflection in the origin in R²: T(x, y) = (−x, −y), V = = (3, 4). 12. T is the reflection in the line y = x in R²: T(x, y) = (y, x), V = (3, 4). 13. T is the reflection in the y-axis in R²: T(x, y) = (−x, y), V = (2, -3). 14. T is the reflection in the x-axis in R²: T(x, y) = (x, −y), V = (4, − 1). 15. T is the counterclockwise rotation of 45° in R², V = (2, 2). 16. T is the counterclockwise rotation of 120° in R², V = (2, 2). 17. T is the clockwise rotation (0 is negative) of 60° in R², V = (1, 2). 18. T is the clockwise rotation (0 is negative) of 30° in R², V = (2, 1). 19. T is the reflection in the xy-coordinate plane in R³: T(x, y, z) = (x, y, − z), v = (3, 2, 2). 20. T is the reflection in the yz-coordinate plane in R³: T(x, y, z) = (− x, y, z), v = = (2, 3, 4). 21. T is the projection onto the vector w = (3, 1) in R²: T(v) = projv, v = (1, 4). 22. T is the reflection in the vector w = (3, 1) in R²: T(v) = 2 projwv – v, v = (1, 4).
Transcribed Image Text:Finding the Standard Matrix and the Image In Exercises 11-22, (a) find the standard matrix A for the linear transformation T, (b) use A to find the image of the vector v, and (c) sketch the graph of v and its image. 11. Tis the reflection in the origin in R²: T(x, y) = (−x, −y), V = = (3, 4). 12. T is the reflection in the line y = x in R²: T(x, y) = (y, x), V = (3, 4). 13. T is the reflection in the y-axis in R²: T(x, y) = (−x, y), V = (2, -3). 14. T is the reflection in the x-axis in R²: T(x, y) = (x, −y), V = (4, − 1). 15. T is the counterclockwise rotation of 45° in R², V = (2, 2). 16. T is the counterclockwise rotation of 120° in R², V = (2, 2). 17. T is the clockwise rotation (0 is negative) of 60° in R², V = (1, 2). 18. T is the clockwise rotation (0 is negative) of 30° in R², V = (2, 1). 19. T is the reflection in the xy-coordinate plane in R³: T(x, y, z) = (x, y, − z), v = (3, 2, 2). 20. T is the reflection in the yz-coordinate plane in R³: T(x, y, z) = (− x, y, z), v = = (2, 3, 4). 21. T is the projection onto the vector w = (3, 1) in R²: T(v) = projv, v = (1, 4). 22. T is the reflection in the vector w = (3, 1) in R²: T(v) = 2 projwv – v, v = (1, 4).
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