As of 29Jan2007:

These are the Extra Problems for Prof. King's Linear Algebra course.

They are typically harder than text problems, and are a chance for you to
show-off.  You are welcome to work in groups on these problems, and you
may post a soln (or partial soln) as a group.

These problems are not due at any particular time; they are for you to
challenge yourself.


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E1:  Let S be the unit sphere in |R^3.   Given a directed-line, L, through
the origin, and given an angle @ (well, imagine here that "@" is "theta"),
let

	R_{L,@}

mean the transformations which rotates S about L by an angle @; so the
center of S stays fixed.  Call such a rotation a "simple-rotation".

  Now suppose that 

	T  :=  R_77 o R_76 o ... o R_2 o R_1

is a composition of simple-rotations, about different directed-lines, and
by different angles; call such a composition, a "complex-rotation".
(Note: The "77" is for specificity; I really mean an arbitrary composition
of simple-rotations.)

   Must every complex-rotation equal a single simple-rotation?  --please
post a proof or counterexample.
 
  Note:  I will tell you that every complex-rotation equals a composition
  of TWO simple-rotations.     Can you post a geometric argument for this?
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E3:
You have a 2-dimensional sphere, and you must use any coordinate system in
such a way as to describe an arbitrary rotation around 2 axes.
ie: YZY(Z^-1)(Y^-1)

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E4: The N-sphere, let's call it S, is the boundary of the unit ball (the
[N+1]-ball) in |R^{N+1}.

In describing a general "rotation" on S, how 

Note: I'm not sure if you were asking this for a 3-dimensional sphere or an
n-dimensional sphere.  I wrote this for a 3-dimensional sphere.
Find the polynomial formula for the dimension of the rotation group of a
3-dimensional sphere in a 4-dimensional space.  The 3-dimensional sphere is
defined as {(x_1, x_2, x_3, x_4) | x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1^2}

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