Student understanding of linear combinations of eigenvectors

Abstract

To contribute to the sparse educational research on student understanding of eigenspace, we investigated how students reason about linear combinations of eigenvectors. We present results from student reasoning on two written multiple-choice questions with open-ended justifications involving linear combinations of eigenvectors in which the resultant vector is or is not an eigenvector of the matrix. We detail seven themes that analysis of our data revealed regarding student responses. These themes include: determining if a linear combination of eigenvectors satisfies the equation \(A\varvec{x}=\lambda \varvec{x}\); reasoning about a linear combination of eigenvectors belonging to a set of eigenvectors; conflating scalars in a linear combination with eigenvalues; thinking eigenvectors must be linearly independent; and reasoning about the number of eigenspace dimensions for a matrix. In the discussion, we explore how themes sometimes cut across questions and how looking across questions gives insight into individuals’ conceptions of eigenspace. Implications for teaching and future research are also offered.

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Notes

  1. 1.

    “An eigenvector of an \(nxn\) matrix \(A\) is a nonzero vector \(\varvec{x}\) such that \(A\varvec{x}=\lambda \varvec{x}\) for some scalar \(\lambda\). A scalar \(\lambda\) is called an eigenvalue of \(A\) if there is a nontrivial solution \(\varvec{x}\) of \(A\varvec{x}=\lambda \varvec{x}\); such a \(\varvec{x}\) is called an eigenvector corresponding to\(\lambda\)” (Lay et al., 2016, p. 269). The eigenspace of \(A\) corresponding to \(\lambda\) is “the set of all solutions of \(A\varvec{x}=\lambda \varvec{x}\), where \(\lambda\) is an eigenvalue of \(A.\) [It c]onsists of the zero vector and all eigenvectors corresponding to \(\lambda\)” (p. A9).

  2. 2.

    This paper builds from and is an extension of a conference presentation given at the 2018 Research in Undergraduate Mathematics Education Conference (Wawro, Watson, & Zandieh, 2018).

  3. 3.

    “A genetic decomposition is a hypothetical model that describes the mental structures and mechanisms that a student might need to construct in order to learn a specific mathematical concept” (Arnon et al., 2014, p. 27).

  4. 4.

    See Tall (2004) for a more detailed discussion of the Three Worlds of Mathematics (geometric/embodied, symbolic, formal) used as part of the framing in Thomas and Stewart (2011).

  5. 5.

    “Process conception” and “object conception” are constructs from APOS Theory; see, for example, Dubinsky and McDonald (2001) for more information.

  6. 6.

    The main MCE version prompts students to justify their answer to the multiple-choice stem by selecting all pre-made justification statements that support their choice (Watson et al., 2017; Zandieh, Plaxco, Wawro, Rasmussen, Milbourne, & Czeranko 2015).

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant Number DUE-1452889. 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.

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Correspondence to Megan Wawro.

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Wawro, M., Watson, K. & Zandieh, M. Student understanding of linear combinations of eigenvectors. ZDM Mathematics Education 51, 1111–1123 (2019). https://doi.org/10.1007/s11858-018-01022-8

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Keywords

  • Linear algebra
  • Student reasoning
  • Eigenspace
  • Linear combination