ABSTRACT
To translate geometry into algebra requires a coordinate system to be chosen. Unless chosen carefully, there is no reason why it should be well adapted to the geometry that interests us. This means that nice geometry can lead to nasty algebra. This sounds like a problem but it is one we can turn to our advantage: it means that nasty algebra can actually arise from nice geometry. This is the idea behind this chapter. It will be used to help us study equations of degree 2 in two or three variables called, respectively, conics and quadrics. Equations such as these can be handled using the methods of this book, but those of higher degree cannot, so they are a fitting place to end. Not surprisingly, the theme that permeates this book — the relationship between algebra and geometry — will be uppermost: we perform algebraic operations motivated by geometric ideas which we ultimately interpret geometrically. This chapter makes heavy use of Section 8.6 and Section 9.5. All the matrices in this chapter will either be 2×2 or 3×3. Thus when we write that a matrix is an n×n matrix we mean that n = 2 or n = 3.
