An Introduction to Probability and Statistical Inference, 2nd Edition

6.3 Linear Transformations

In this section, a brief discussion is presented for a specific kind of transformation, linear transformations. The basic concepts and results used here can be found in any textbook on linear algebra.

Definition 3

Suppose the variables x₁,…,x_k are transformed into the variables y₁,…, y_k in the following manner:

$y_i={\displaystyle \sum _{j=1}^k{c}_{ij}{x}_j,\text{?}\text{?}\text{?}i=1,\dots ,k,}$

  (14)

where the c_ij's are real constants. Such a transformation is called a linear transformation (all the x_i's enter into the transformation in a linear way, in the first power).

Some terminology and elementary facts from matrix algebra will be used here. Denote by C the k × k