Our main purpose in this chapter and the next is to show what matrices are good for; but before we can use this powerful tool we must understand how it works. We shall therefore begin by constructing it, starting with familiar pieces and showing, along the way, new efficient ways of looking at and talking about geometry, trigonometry, and their merger with algebra. In Chapter 5 we aim to show how this facility, used with bold imagination, devastates comfortable, commonplace conceptions of our physical world.

Why Use Matrices?

Have you ever tried using a lump of rock to drive a six-inch nail into a four-inch beam? It is easier with a hammer. Easier, because the hammer is designed expressly for the job, designed to have good balance, to handle well, to effect a neater job with less effort. Its design, deceptively simple, is the result of much thought about questions of rigidity, distribution of weight and center of percussion. Hard thinking goes into its design; hard work is eased by its use.

Matrices, too, are deceptively simple. Clever people have given much thought to devising a notation that handles well and a technique that does a tidier, more effortless job. Yes, matrices take the slog out of nailing equations. And, as with driving nails, there is no need to take anyone's word for it; experience is conclusive.