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)