07-401/Homework Assignment 9

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Reading

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.

Doing

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 (#).

Due Date

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

Solutions

Just for Fun

1. 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.
2. Prove that $O(3)$ is not solvable (though note that the similarly-defined group $O(2)$ is solvable).