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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