Practical Linear Algebra, 3rd Edition

Chapter 4

Changing Shapes: Linear Maps in 2D

Geometry always has two parts to it: one part is the description of the objects that can be generated; the other investigates how these objects can be changed (or transformed). Any object formed by several vectors may be mapped to an arbitrarily bizarre curved or distorted object—here, we are interested in those maps that map 2D vectors to 2D vectors and are “benign” in some well-defined sense. All these maps may be described using the tools of matrix operations, or linear maps. An interesting pattern is generated from a simple square in Figure 4.1 by such “benign” 2D linear maps—rotations and scalings.

Figure 4.1

Linear maps in 2D: an interesting geometric figure constructed by applying 2D linear ...