Geometry of Linear Maps

Abstract

The geometric meaning of a linear function \(x \mapsto y = mx\) is simple and clear: it maps \(\mathbb{R}^1\)to itself, multiplying lengths by the factor m. As we show, linear maps \(M:\mathbb{R}^n\to\mathbb{R}^n\) also have their multiplication factors of various sorts, for any n > 1. In later chapters, these factors play a role in transforming the differentials in multiple integrals that is exactly like the role played by the multiplier φ'(s) in the transformation dx = φ'(s)ds in single-variable integrals.With this in mind, we take up the geometry of linear maps in the simplest case of two variables.