Show that this 22

definitionisthesameasSU(2)= :,C,|| +|| =1 .

By writing = x1 +ix2, = x3 +ix4 where xi R, show that SU(2) is diffeomorphic1

tothe3-sphereS3 ={(x1,x2,x3,x4)R4 :x21+x2+x23+x24 =1}.

2. Let A be a 2 matrix with real entries. Show that A O(2) if and only if

cos sin cos sin A= sin cos orA= sin cos .

1 a b

3. Consider the set ofH = 0 1 c a,b,c R

, which we discussed in Exercise 1.

0 0 1

Show that H is a Lie group2. Determine the center of H, Z(H).

4. Consider the matrix valued functions A, B : I Mn(R). What does it mean for A(t) and B(t) to be smooth? Show that if A(t) and B(t) are smooth, then so is A(t)B(t),

and

d[A(t)B(t)]= dAB(t)+A(t)dB. dt dt dt