Math 331 - February 10, 1999

Discussion

• Finished discussion of topics raised previously.

Exercises due Friday, February 12

1. Show that if an affine map of the plane is not a transformation and does not map the entire plane to a single point, then it maps the entire plane to a single line.

2. How many of the 24 permutations of the vertices of a parallelogram may be realized with an isometry when:

   1. the parallelogram is a square?

   2. the parallelogram is a rectangle but not a square?

   3. the parallelogram is non-rectangular?

3. Let f(x) = Ax be the linear transformation of the plane where A is the matrix

   A = {1}/{5} (

   3	-4
   4	3

   ) .

   1. What points x of the plane are ``fixed'' by f, i.e., satisfy f(x) = x ?

   2. What lines in the plane are carried by f to other lines?

   3. What lines L in the plane are ``stabilized'' by f, i.e., satisfy the condition that f(x) is on L if x is on L ?