By Dr. Morton L. Curtis (auth.)

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D. det A E {l,-l}. We define SO(n) {A E ~(n) I I} det A and eall this the special orthogonal group (also ealled the rotation group). Similarly, we define SU(n) (A E I U(n) det A I} and eall this the special unitary group. An example of an element of e l = (1,0) to sends el ~(2) and sends - SO(2) (1 is e 2 = (0,1) o 0) -1 to This It is just the refleetion in the first axis, and has determinant equal to -1 . C. The isomorphism question At the end of Chapter I we showed that two groups whieh were defined quite differently were isomorphie.

Exercises 1. Prove Proposition 1. 2. Prove Proposition 4. 3. Let A be any element of with ~(n) det A -1. Show that = ~(n) - SO(n) 4. Show that any element of (cos 8 -sin 8 5. A E If A E and U(n) ).. E C [BA I B SO(2) E SO(n)} . can be written as sin 8) cos 8 has length one, show that U(n) 6. Let LI that reflection in and L2 LI be lines through the origin in followed by reflection in L2 and L2 rotation through twice the angle between 7. A matrix A E Show that the image of set of A Mn(F) ~n LI 2 Show equals a is said to be idempotent if under P AA = A .

The general 2 x2 is ~ ~(n) real skew-sYUlUletrie matrix is of the form 2 48 0. - To ca1cu1ate 0. 3 -(:3 _:3) eCl. _ e"' , CI. o x x e: R • (-x 0)' we ca1cu1ate the powers of 4 - (': X:) CI. , (0 5 X') - 1 +2T X) 0 (-: -x 2 0. • 0. 2 - (-: _:3 ) 0) 1 ( 0 2 + 3T 3 -x x 1 x4 0 ) S 1 (0 5 x ) + ... + . ( 0 x 4 +sr -x 0 (10)+(0 o 1 -x _:2) Then etc. •• - cos x , etc. We find that eCl. _ ( cos x -sin x which is a plane rotation of skew-symmetric matrix x Bin x) cos x radians . Thus for any real 2)( 2 we have CI.