Chapter 2 - Space and Linearity

Abstract

The chapter presents the analysis of signals and their transformations with an overview of the applicable linear algebra followed by the results from advanced calculus needed to understand infinite dimensional spaces. The ability to see geometric properties of objects in space helps to visualize important properties of digital signals. However, it is necessary to find correspondence between the signal property and the geometric object. A starting point is the analytic geometry of the line, the plane, and the space. Digital signals representing sounds and images are modeled by points in some of these generalized spaces, and many common transformations of such signals are easily described as geometric operations on those points. For example, points in space may be added together or multiplied by real numbers that correspond respectively to mixing signals or amplify them. The results are variously called linear combinations, superpositions, or linear transformations. Linear transformations of linear spaces preserve linear combinations. Even rather complicated transformations satisfy this linearity assumption, which in many cases reduces their analysis to linear algebra.