## ABSTRACT

A fundamental theme of many branches of mathematics is the study of func‐ tions, or transformations, between vector spaces. A function can be first clas‐ sified as linear or nonlinear. For example, f : R → R ${\mathbb{R}} \to {\mathbb{R}}$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/inline-math8_1.tif"/> with f(x) = 2x is a linear function, while g : R → R ${\mathbb{R}} \to {\mathbb{R}}$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/inline-math8_2.tif"/> with g(x) = x 2 is a nonlinear function. A linear function can be further classified as homogeneous linear or nonhomogeneous linear. For example, f : R → R ${\mathbb{R}} \to {\mathbb{R}}$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/inline-math8_3.tif"/> with f(x) = 2x is homogeneous linear while h : R → R $h:{\mathbb{R}} \to {\mathbb{R}}$ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315172309/e112b15c-ee8b-4102-9c2c-0d5caba3f84a/content/inline-math8_4.tif"/> with h(x) = 2x + 1 is nonhomogeneous linear. As an important application of matrix theory, we devote this chapter to a brief discussion of homogeneous linear functions between vector spaces. In what follows, if no confusion otherwise arises, we say that a function is linear when it is homogeneous linear.