Chapter 4 - Linear Transformations from a Geometric Viewpoint

Abstract

This chapter deals with linear transformations from a geometric viewpoint. This chapter describes the geometric aspects of various kinds of matrix transformations. This chapter interestingly discusses that how general linear transformations can be viewed as composites of simple kinds of matrices that, individually, are more easily portrayed from a geometric viewpoint. This chapter starts out with an overview description of matrix transformations and their representation as sets of linear equations. It focuses on the geometric representation of various types of matrix transformations, such as rotations, stretches, central dilations, and reflections. The geometric character of combinations of various matrix transformations is also illustrated so that the reader can see how simple geometric changes, when taken in combination, lead to complex representations. This chapter elaborates on the solution of simultaneous equations and the central roles that matrix inversion and matrix rank play in this activity. Finally, this chapter discusses the solution of linear equations in multivariate analysis.