07-401/Homework Assignment 9

Read chapter 32 of Gallian's book three times:

First time as if you were reading a novel - quickly and without too much attention to detail, just to learn what the main keywords and concepts and goals are.
Second time like you were studying for an exam on the subject - slowly and not skipping anything, verifying every little detail.
And then a third time, again at a quicker pace, to remind yourself of the bigger picture all those little details are there to paint.

Solve problems 22#, 23, 24#, 25, 26# and 27# in Chapter 32 of Gallian's book but submit only the solutions of the problems marked with a sharp (#).

This assignment is due in class on Wednesday April 4, 2007.

Explain how the group $O(3$ ) of rigid rotations of ${\mathbb {R} }^{3}$ can be identified with the subgroup $\{A\in M_{3\times 3}({\mathbb {R} }):\,A^{T}A=I\}$ of the group of invertible $3\times 3$ matrices.
Prove that $O(3)$ is not solvable (though note that the similarly-defined group $O(2)$ is solvable).

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