Vector Spaces or Linear Spaces

Abstract

The introduction of Cartesian coordinates in the 1630s by the French mathematicians Pierre de Fermat (1601–1665) and René Descartes (1596–1650) had a profound influence on the development of mathematics. Cartesian coordinates allowed mathematicians to introduce algebraic methods into the study of geometry. By the middle of the 1800s, however, there was a general dissatisfaction with these coordinate methods and mathematicians began to look for more general and abstract ways by which they could study geometry without referring to coordinate systems. This search leads to the development of vectors and matrices, and eventually to the development of vector spaces or linear spaces.

Among the first to introduce a geometry without coordinates was the Prussian mathematician Hermann Günter Grassman (1809–1877). In his 1844 book Die Ausdehnungslehre (Extension Theory) he introduced the notion of abstract quantities and he defined the operations of addition and scalar multiplication on these abstract quantities. It was, however, the Italian mathematician Giuseppe Piano (1858–1932) who introduced the first axiomatic definition of a vector space in his 1888 book Calcolo Geometrico (Geometric Calculus). This book was well in advance of its time and introduced most of the concepts of vector spaces. The complete axiomatic definition of vector spaces appeared in the 1920 doctoral dissertation of the Polish mathematician Stefan Banach (1892–1945).

As the theory of vector spaces underpins much of the work in computer graphics and curve and surface theory, we will briefly introduce some of the basic ideas of vector spaces in this chapter.

Underlying the concept of a vector space is the concept of a scalar field. Let us start by defining what a scalar field is.