Difference between revisions of "07-401/Homework Assignment 9"

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

Revision as of 22:24, 28 March 2007


Contents

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.

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^TA=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).