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Problem 2.5. In each of the following, an affine transformation F: R² → R² is described. Find a 2 × 2 matrix A and a vector 6 so that F(x) = A·x+b. (a) F is the reflection along the line x = 1. Hint: First translate the line x = 1 to the y-axis by the translation → - 7, then reflect in the y-axis, then translate the y-axis back to the line x = 1. (b) F is the reflection along the line x - y = 2. Hint: Note that the line passes through the point (1, -1), so translate it to the line x = y, then reflect and then translate back. (c) F is the counterclockwise rotation through an angle of 90° about the point (2,3). Hint: Translate the point (2,3) to the origin, then rotate through 90°, then translate back.

Problem 2.5. In each of the following, an affine transformation F: R² → R² is described. Find a 2 × 2 matrix A and a vector 6 so that F(x) = A·x+b. (a) F is the reflection along the line x = 1. Hint: First translate the line x = 1 to the y-axis by the translation → - 7, then reflect in the y-axis, then translate the y-axis back to the line x = 1. (b) F is the reflection along the line x - y = 2. Hint: Note that the line passes through the point (1, -1), so translate it to the line x = y, then reflect and then translate back. (c) F is the counterclockwise rotation through an angle of 90° about the point (2,3). Hint: Translate the point (2,3) to the origin, then rotate through 90°, then translate back.

Advanced Engineering Mathematics
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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Problem 2.5. In each of the following, an affine transformation F: R² → R² is described. Find a 2×2 matrix A and a vector b so that F(x) = A·x+b. (a) F is the reflection along the line x = 1. Hint: First translate the line x = 1 to the y-axis by the translation ☀ → ☎ – 7, then reflect in the y-axis, then translate the y-axis back to the line x = 1. (b) F is the reflection along the line x − y = 2. Hint: Note that the line passes through the point (1, − 1), so translate it to the line x = y, then reflect and then translate back. (c) F is the counterclockwise rotation through an angle of 90° about the point (2,3). Hint: Translate the point (2,3) to the origin, then rotate through 90°, then translate back.
Transcribed Image Text:Problem 2.5. In each of the following, an affine transformation F: R² → R² is described. Find a 2×2 matrix A and a vector b so that F(x) = A·x+b. (a) F is the reflection along the line x = 1. Hint: First translate the line x = 1 to the y-axis by the translation ☀ → ☎ – 7, then reflect in the y-axis, then translate the y-axis back to the line x = 1. (b) F is the reflection along the line x − y = 2. Hint: Note that the line passes through the point (1, − 1), so translate it to the line x = y, then reflect and then translate back. (c) F is the counterclockwise rotation through an angle of 90° about the point (2,3). Hint: Translate the point (2,3) to the origin, then rotate through 90°, then translate back.
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