Linear algebra is one of the unavoidable advanced courses that many mathematics students encounter at university level. The research reported here was part of the first author's recent PhD study, where she created and applied a theoretical framework combining the strengths of two major mathematics education theories in order to investigate the learning and teaching of basic linear algebra concepts. This article highlights some of the overall findings of this research with regard to student understanding of these concepts, and suggests applications of the framework for learning and teaching in undergraduate mathematics classrooms. (Contains 7 figures.)