notes for Mon 2/14
  * homework
     * isomorphism is an equivalence relation
     * equivalence => disjoint sets
     * "conjugacy" satisfies equivalence relation

  * examples of relations which is _not_ an equivalance?

  * is "even" an equivalence relation?  (what are disjoint sets)

  * MATRICES:
     * multiplication
     * inverses
     * determinants ( if = 0, not invertable )
     * transpose    ( swap rows and columns )
     * identify matrix
     * orthogonal iff transpose = inverse

  * matrices "act on" vectors

  * matrix groups (chap 9 in Armstrong)
     * GLn(R)  : "general linear group of invertable n x n matrices", real entries
     * On   : subgroup of GLn that are orthogonal
     * SOn  : subgroup of On that have determinant = +1
     * rotation matrices in 2D and 3D
     * spacial inversion matrices in 3D


assign:

    1.Prove that the "solid cube group" is isomorphic to S4. (See Armstrong, chap 8) 
    2.Practice some matrix operations if you need to: multiplication, determinant, 
      inverse, as described in class and/or in any standard calculus-level text. 
    3.Armstrong, 9.12: Prove that these four matrices form a subgroup of SO3, and 
      find the corresponding rotations.

                      1 0 0    1  0  0    -1 0  0    -1  0 0 
                      0 1 0    0 -1  0     0 1  0     0 -1 0
                      0 0 1    0  0 -1     0 0 -1     0  0 1

     4.Armstrong 9.8: Show that these two matrices represent rotations, 
       and find the angle and axis for each: 
          2/3   1/3  2/3     |      -1/sqrt(2)    1/sqrt(3)   1/sqrt(6)
         -2/3   2/3  1/3     |       1/sqrt(2)    1/sqrt(3)   1/sqrt(6)
         -1/3  -2/3  2/3     |       0            1/sqrt(3)  -2/sqrt(6)