Is it a subspace or not? Making sense of subspaces of vector spaces in a technology-assisted learning environment

Abstract

The purpose of this research study was to understand how linear algebra students in a university in the United States make sense of subspaces of vector spaces in a series of in-depth qualitative interviews in a technology-assisted learning environment. Fourteen mathematics majors came up with a diversity of innovative and creative ways in which they coordinated visual and analytic approaches in the exploration of subspaces of a number of vector spaces such as ${\mathbb{R}}^n ,{\mathbb{R}}^{n \times n} ,{\mathbb{P}}_n$ and $C\left( {\mathbb{R}} \right)$. In the demonstration of the closure property, the research participants embraced three different approaches in expressing the linear combination of ${\mathbf{u}}$ and ${\mathbf{v}}$: (i) ${\mathbf{u}} + {\mathbf{v}}$ [type I notation]; (ii) $c{\mathbf{u}} + {\mathbf{v}}$ [type II notation]; and (iii) $c{\mathbf{u}} + d{\mathbf{v}}$ [type III notation]. Although the majority of the students embraced type I notation at the beginning; there was a considerable shift towards type II notation towards the end of the study. Although students’ visualizations on the DGS appeared at times to be in agreement with the vector sum notation analytically embraced, students still most of the time utilized the type I notation in their visualizations. The role of the zero vector manifested itself in three main categories: (i) to show nonemptiness of a subspace; (ii) to show the failure of closure property of a nonsubspace [e.g., ${\mathbf{u}},{\mathbf{v}} \in W,{\text{but}}\,{\mathbf{u}} + {\mathbf{v}} = {\mathbf{0}} \notin W$]; (iii) to show nonsubspaceness by showing that $0 \notin W$. The zero vector seemed to be emphasized both within the visual approach and the analytic approach. Moreover, the visual approach did not prove necessary except for the nonsubspace explorations of certain cases. The paper concludes by offering pedagogical implications along with implications for the mathematics teaching profession and recommendations for further research.