Reasoning About Solutions in Linear Algebra: the Case of Abraham and the Invertible Matrix Theorem

Abstract

A rich understanding of key ideas in linear algebra is fundamental to student success in undergraduate mathematics. Many of these fundamental concepts are connected through the notion of equivalence in the Invertible Matrix Theorem (IMT). The focus of this paper is the ways in which one student, Abraham, reasoned about solutions to A x = 0 and A x = b to draw connections between other concept statements within the IMT. Data sources were video and transcripts from whole class discussion, small group work, and individual interviews. The overarching analytical structure was influenced by a framework of genetic analysis (Saxe, Journal of the Learning Sciences, 11, 275–300, 2002), and Toulmin’s Model of Argumentation (1969) was employed to analyze the structure of arguments both in isolation (microgenesis) and over time (ontogenesis). This case study, rather than focusing on student difficulties in undergraduate mathematics, serves as a compelling example of the productive and powerful reasoning that is possible as students make sense of complex mathematics. The results present an ontogenetic analysis of Abraham’s use of solutions to reason about how span and linear independence of a set of vectors are related, as well microgenetic analyses of various examples of reasoning about solutions to justify connections between other concepts within the IMT.