Linear Mappings

Abstract

Although the majority of the topics in this book (all the topics taken into account excluding only complex polynomials) are related to linear algebra, the subject “linear algebra” has never been introduced in the previous chapters. More specifically, while the origin of the term algebra has been mentioned in Chap. 1, the use of the adjective linear has never been discussed. Before entering into the formal definitions of linearity, let us illustrate the subject at the intuitive level. Linear algebra can be seen as a subject that studies vectors. If we consider that vector spaces are still vectors endowed with composition laws, that matrices are collections of row (or column) vectors, that systems of linear equations are vector equations, and that a number can be interpreted as a single element vector, we see that the concept of vector is the elementary entity of linear algebra. As seen in Chap. 4, a vector is generated by a segment of a line. Hence, the subject linear algebra studies “portions” of lines and their interactions, which justifies the adjective “linear”.