See also the discussion of February 22
A rotation matrix is a 2 \times 2 orthogonal matrix of determinant 1. It always has the form
| cos t | - sin t |
| sin t | cos t |
) .
A reflection matrix is a 2 \times 2 orthogonal matrix of determinant -1. It always has the form
| cos t | sin t |
| sin t | - cos t |
) .
The number in each box indicates the number of parameters for that type of isometry.
| orientation-preserving | orientation-reversing | |
| fixed points | rotations (3) | reflections (2) |
| no fixed points | translations (2) | glide reflections (3) |
An isometry is orientation-reversing if and only if it has the form f(x) = U x + b where U is a reflection matrix. In this case let
Then one has the decomposition
of f as the reflection sigma (since b' is in the image of 1 - U -- see the discussion of February 22) followed by the translation tau_{b''}, a translation that is parallel to the axis of sigma. Thus, f is a reflection if and only if (1 + U)b = 0 since then the translation by the vector b'' is the identity map. Otherwise f is a glide reflection.
Whether f is a reflection or a glide reflection, it has an axis that is the axis of the reflection sigma.
Note that the parallel class of the axis of f depends only on U and not on b. In fact the axis of f is always parallel to the eigenspace of U for the eigenvalue 1, which is the same thing as the null space of the matrix 1 - U.
Note further that the null space of 1 - U is the image of mulitplication by 1 + U, and ``dually'' the null space of 1 + U is the image of multiplication by 1 - U. In fact,
give rise to linear maps that are known in linear algebra as orthogonal projections. One has the relations p^{2} = p, q^{2} = q, pq = 0, qp = 0, p + q = 1, and q - p = U.
Study all of the material above.
Catch up on previous exercise sets.
Remember that there will be a midterm test on Monday, March 22.