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Problem 3. Let GL2 (R) denote the subset of M₂(R) consisting of invertible matrices this is called the general linear group of degree 2. Given A E GL₂(R) and b = R², we define TA,b: R² → R² by TA(v) = Av+b VvER². Recall that the set of all such transformations is denoted Aff2(R) and called the group of affine transformations of the plane. 3.1. Show that there is a matrix CE GL2 (R) and a vector d = R² such that TA,b TC,d=TA,b= TC,d° TA,b for all TA, Aff2 (R). 3.2. Let Tab € Aff₂ (R). Find a matrix C = GL₂(R) and a vector d = R² such that (TA,b)-¹ = TC,d- 3.3. Let TA, and TB, be two fixed affine transformations. Find a matrix C E GL2(R) and a vector de R² such that TA,b° TB,C = TC.d

Problem 3. Let GL2 (R) denote the subset of M₂(R) consisting of invertible matrices this is called the general linear group of degree 2. Given A E GL₂(R) and b = R², we define TA,b: R² → R² by TA(v) = Av+b VvER². Recall that the set of all such transformations is denoted Aff2(R) and called the group of affine transformations of the plane. 3.1. Show that there is a matrix CE GL2 (R) and a vector d = R² such that TA,b TC,d=TA,b= TC,d° TA,b for all TA, Aff2 (R). 3.2. Let Tab € Aff₂ (R). Find a matrix C = GL₂(R) and a vector d = R² such that (TA,b)-¹ = TC,d- 3.3. Let TA, and TB, be two fixed affine transformations. Find a matrix C E GL2(R) and a vector de R² such that TA,b° TB,C = TC.d

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Problem 3. Let GL2 (R) denote the subset of M₂(R) consisting of invertible matrices this is called the general linear group of degree 2. Given A E GL₂(R) and b = R², we define TA,b: R² → R² by TA(v) = Av+b VvER². Recall that the set of all such transformations is denoted Aff2(R) and called the group of affine transformations of the plane. 3.1. Show that there is a matrix CE GL2 (R) and a vector d = R² such that TA,b TC,d=TA,b= TC,d° TA,b for all TA, Aff2 (R). 3.2. Let Tab € Aff₂ (R). Find a matrix C = GL₂(R) and a vector d = R² such that (TA,b)-¹ = TC,d- 3.3. Let TA, and TB, be two fixed affine transformations. Find a matrix C E GL2(R) and a vector de R² such that TA,b° TB,C = TC.d
Transcribed Image Text:Problem 3. Let GL2 (R) denote the subset of M₂(R) consisting of invertible matrices this is called the general linear group of degree 2. Given A E GL₂(R) and b = R², we define TA,b: R² → R² by TA(v) = Av+b VvER². Recall that the set of all such transformations is denoted Aff2(R) and called the group of affine transformations of the plane. 3.1. Show that there is a matrix CE GL2 (R) and a vector d = R² such that TA,b TC,d=TA,b= TC,d° TA,b for all TA, Aff2 (R). 3.2. Let Tab € Aff₂ (R). Find a matrix C = GL₂(R) and a vector d = R² such that (TA,b)-¹ = TC,d- 3.3. Let TA, and TB, be two fixed affine transformations. Find a matrix C E GL2(R) and a vector de R² such that TA,b° TB,C = TC.d
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