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

Let T be a linear transformation from M2,2 into M2,2 such that T([1000])=[1102], T([0100])=[0211], T([0010])=[1201],T([0001])=[3110]. Find T([1314]).

The gray square in Table 1 has the following vertices: [00],[10],[11],[01] Apply each of the three transformations given in Table 1 to these vertices and sketch the result to verify that each transformation has the indicated effect. Use c=2 in the expansion matrix and c=1 in the shear matrix.

Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G. Sec. 3.3,22b,32b Find the center Z(G) for each of the following groups G. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1. Find the centralizer for each element a in each of the following groups. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 Sec. 4.1,22 22. Find an isomorphism from the octic group D4 in Example 12 of this section to the group G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of Section 3.1. Sec. 4.6,14 14. Let G={ I2,R,R2,R3,H,D,V,T } be the multiplicative group of matrices in Exercise 36 of section 3.1, let G={ 1,1 } under multiplication, and define :GG by ([ abcd ])=adbc. Assume that is an epimorphism, and find the elements of K= ker . Write out the distinct elements of G/K. Let :G/KG be the isomorphism described in the proof of Theorem 4.27, and write out the values of .