Linearity

Abstract

We began this book by learning how to systematically solve systems of linear algebraic equations. This “elementary” problem formed our launching pad for developing the fundamentals of linear algebra. In its initial form, matrices and vectors were the primary focus of our study, but the theory was developed in a sufficiently general and abstract form that it can be immediately used in many other useful situations — particularly infinitedimensional function spaces. Indeed, applied mathematics deals, not just with algebraic equations, but also with differential equations, difference equations, integral equations, stochastic systems, differential delay equations, control systems, and many other types — only a few of which, unfortunately, can be adequately developed in this introductory text. It is now time to assemble what we have learned about linear algebraic systems and place the results in a suitably general framework that will lead to insight into the key principles that govern all linear systems arising in mathematics and its applications.