CHAPTER 4 - Linear Transformations from a Geometric Viewpoint

Abstract

This chapter discusses linear transformations from a geometric viewpoint and describes the geometric aspects of various kinds of matrix transformations. Mapping is an operation by which elements of one set of mathematical entities are transformed into elements of another. Values obtained by a mapping are called images, while the values being transformed are often called preimages of the transformation. Any linear transformation can be represented in matrix form. The need for matrix transformation arises naturally in the solution of linear equations. A regular inverse of the matrix A is denoted by A-1, and, when it exists, it is unique. The simplest type of transformation is represented by a point transformation. Many matrix operations can be usefully represented geometrically if two or three dimensions are involved. Translations are frequently used in multivariate analysis. Permutation of a set of points involves a matrix transformation that carries each coordinate value on axis 1 into a corresponding coordinate on axis 2, and vice versa.