Question

Problem B. Define a function f: C GL2(R) by the following formula f(a+ib) = () a-b 1 (a) Check that f is a homomorphism. Is f injective? Is f surjective? (b) Verify that f takes the complex unit circle C into the group SO2(R) of rotation matrices (ossin) Prove that the resulting map sin cos f: C SO2(R) is an isomorphism. List all matrices of the form f(z) for some z E U3

U3 is the notation for the group of 3rd roots of untity— U3={ a complex number z : z^3=1}

Answer 1

Q GL₂(IR) by the formula solution Define a function of: f(arib) = Let Zizatib and Z2e = @tid be two elements in cho Now Z1 Z2 = catib) (eeld) (ac-bd) tilbetad) if(zzz) = (ac-bd -(berad)) betad ae-bd fis a group homomornom from cx into ELL(R) kerf={zzaribe c*: f(2)=(181) = {z=artib: (ana) = (01) - {zzatib' 221, 6205 243. fan injective clearply. (10) EGLE(R) but (u) has no preimaze in ax undersf. so fîn not surfective c = {ZEC: 121=18 any element in It has polars from eit=cs&tising Define fi 8 5 0(R) by where SO₂(OR) lloso ofreio) = rotation matrix -sine sino le group in R? ( 1038 -sivos sino aso) S so if we tarke. a=colo, b=sino. them flere) = f(aso risine) (sine aso kerner of f= Lelofc': fleio)= (os. = { CBD ti sine ect: ( no suo ) - (i)} {asotisine EC' C30=1 and sinoor in fin injective and every element of soL(R) the form aso -sino felo) Sino caso fissurfective this finan isomorphium-

let Uz = group of 3rd roots of unity - ZEC: z3=14 Wahltasi 2 وحی الہار لما ۱٫ د Whz - 1 - Noi f(1) = (18) f(w) = ( 2 - 23) List of maprices flz), zt3 n - Nh N NOW \ -V. frwn) = (1 2 3 - Z