Generating Exotic-Looking Vector Spaces

Abstract

Many introductory textbooks on linear algebra introduce the definition of a vector space using abstract notation for vector addition and scalar multiplication (such as ? and 0, respectively), but then they generally limit the exercises and examples to classic problems where vector addition and scalar multiplication are those defined in Rn. This note describes how to generate computational exercises designed for teaching students the axioms of vector spaces using nonstandard operations for vector addition and scalar multiplication. Such exercises have the pedagogical value of allowing the student to study the axioms of vector spaces using familiar objects, such as real numbers, but with unfamiliar operations for vector addition and scalar multiplication. Checking the vector space axioms in such exotic vector spaces helps students develop a deeper understanding of these axioms. The basis for generating these exercises lies in the following theorem.