ABSTRACT
The foundation for linear algebra is the notion of a vector space over a field. Two operations are important in a vector space (i) addition: any two elements in a vector space can be added together; (ii) multiplication by a scalar: an element in a vector space can be multiplied by a scalar (= an element of the field). Anytime one has mathematical objects where these two operations are well-defined and satisfy some basic properties, one has a vector space. Allowing this generality and developing a theory that just uses these basic rules, leads to results that can be applied in many settings.
