## M4005 Geometry

### March 20, 2006

Yet more discussion of coordinate systems, with the goal of showing that in any affine coordinate system every straight line is the set of solutions to a linear equation and the set of solutions to any linear equation is a straight line. We call a coordiante system affine if it is EITHER a standard coordinate system with perpendicular axes and the same unit of distance on each OR what we have called a skew coordinate system. By "linear equation" we mean an equation in the variables x and y that can be put in the form Ax + By + C = 0, where A, B, and C are constants and either A or B (or both) are nonzero.

This class also included a discussion (45 minutes) on functions. Key concepts included:

• Vocabulary: domain, codomain, injection ("one-to-one" function), surjection ("onto" function), bijection (a function that is both injective and surjective).
• The idea that a coordiante system on the plane is a pair of functions.

Second handout on coordinates.