In Chapter 1, we developed the theory of abstract vector spaces in considerable detail. It is now natural to consider those functions defined on vector spaces that in some sense “preserve” the structure. These special functions are called linear transformations, and they abound in both pure and applied mathematics. In calculus, the operations of differentiation and integration provide us with two of the most important examples of linear transformations (see Examples 6 and 7 of Section 2.1). These two examples allow us to reformulate many of the problems in differential and integral equations in terms of linear transformations on particular vector spaces (see Sections 2.7 and 5.2).

In geometry, rotations, reflections, and projections (see Examples 2, 3, and 4 of Section 2.1) provide us with another class of linear transformations. Later we use these transformations to study rigid motions in ${\text{R}}^n$ (Section 6.10).

In the remaining chapters, we see further examples of linear transformations occurring in both the physical and the social sciences. Throughout this chapter, we assume that all vector spaces are over a common field F.