1. (a) Write down, as a linear function of a single complex number, the
       transformation that rotates the plane 90 degrees counterclockwise
       around the point (1,0).

   (b) Write down the same transformation as a 3x3 matrix acting on the column
       vector (x,y,1).

2. (a) Write pseudocode for an algorithm that takes as input the projective
       coordinates of three lines and returns true if the three lines go
       through a single point of the projective plane, and false if they do
       not. You may assume the existence of a subroutine for computing
       the determinants of 3x3 matrices.
    
   (b) Do the three lines [0,1,1], [-1,0,1], and [1,1,0] go through a single
       point of the projective plane? If so, what is that point?

3. Any two points p and q of the projective plane determine a line pq. Points p and q partition the points on pq into two subsets: the subset between p and q, and another subset that contains a point at infinity. The subset between p and q is a line segment.

In the definition of a line segment above, substitute points for lines and lines for points to determine the kind of geometric object that is dual to a line segment. What kind of object is it? Hint: if we use the duality transformation described in the lecture, that preserves vertical distance, the dual of a point at infinity is a vertical line.