Varieties in the use of geometry in the teaching of linear algebra

Geometry is used in different ways in the teaching of linear algebra. In this paper, I offer a typology of these ways, which I call varieties, and address three central questions. The first question is, What varieties of use of geometry in the teaching of linear algebra exist? This question is addressed through an analysis of six linear algebra textbooks, republished in multiple editions in the last decade or so. The analysis resulted in seven varieties, which can be used by researchers to investigate systematically the use of geometry in the teaching of linear algebra, in and outside the classroom. The second question is, What are the different impacts of these varieties on students’ ability to accomplish the following: (a) abstract geometrically-based linear-algebraic concepts into general representations? (b) Extend geometrically-based linear-algebraic concepts to their counterparts in other models, such as space of polynomials, functions, and matrices? This question is addressed through analyses of a sample of various results reported in the literature. The third question is, What might account for these impacts? A conceptual basis underlying the results discussed in this paper is theorized.