Math 403: Euclidean Geometry
Final Exam - Review

Time & Location: Friday, May 6, 1:30-4:30, 441 Altgeld.
Material: The final exam will be cumulative. This review sheet concerns only the material covered in the few lectures since the third exam. Feel free to consult the first three review sheets to review the material from earlier in the course.

You should be prepared to state any of the following

Definitions:

• Orientation-preserving isometry
• Orientation-reversing isometry
• Matrix corresponding to a linear map
• Trace of a linear map or matrix
• Determinant of a linear map or matrix
• General linear group, orthogonal group
• Wallpaper groups

Results:

• Conjugation formula for rotations, depending on orientations
• Conjugation formula for translations by linear isometries
• The structure theorem for finite groups of isometries (Theorem 4.45)
• Composition of linear maps corresponds to matrix multiplication
• Change of basis formula
• Possible rotations in a wallpaper group

In addition, you should know how to write down a matrix corresponding to a linear map, given a choice of basis. You should also understand how to apply the change-of-basis formula to obtain the matrix with respect to a new basis.

Suggested Exercises: Try the odd-numbered exercises in Chapter 5. Some of these were done in class, and they all have hints in the back of the book, but try to do them without consulting your notes or the back of the book. Another good thing to try is to look at some of the results in the text (like Proposition 4.41 or Theorem 5.1, for example) and try to prove them without looking at the proof in the book.