# Subspace in linear algebra: investigating students’ concept images and interactions with the formal definition

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## Abstract

This paper reports on a study investigating students’ ways of conceptualizing key ideas in linear algebra, with the particular results presented here focusing on student interactions with the notion of subspace. In interviews conducted with eight undergraduates, we found students’ initial descriptions of subspace often varied substantially from the language of the concept’s formal definition, which is very algebraic in nature. This is consistent with literature in other mathematical content domains that indicates that a learner’s primary understanding of a concept is not necessarily informed by that concept’s formal definition. We used the analytical tools of concept image and concept definition of Tall and Vinner (Educational Studies in Mathematics, 12(2):151–169, 1981) in order to highlight this distinction in student responses. Through grounded analysis, we identified recurring concept imagery that students provided for subspace, namely, geometric object, part of whole, and algebraic object. We also present results regarding the coordination between students’ concept image and how they interpret the formal definition, situations in which students recognized a need for the formal definition, and qualities of subspace that students noted were consequences of the formal definition. Furthermore, we found that all students interviewed expressed, to some extent, the technically inaccurate “nested subspace” conception that **R** ^{ k } is a subspace of **R** ^{ n } for *k* < *n*. We conclude with a discussion of this and how it may be leveraged to inform teaching in a productive, student-centered manner.

## Keywords

Linear algebra Subspace Concept image and concept definition Undergraduate mathematics education## Notes

### Acknowledgments

This material is based upon work supported by the National Science Foundation under grants no. DRL 0634099 and DRL 0634074. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. We are grateful to Chris Rasmussen and Michelle Zandieh for helpful advice during data collection and analysis.

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