# 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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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).

## References

1. Arnon, I., Cottrill, J., Dubinsky, E., Oktac, A., Roa Fuentes, S., Trigueros, M., et al. (2014). APOS Theory. A framework for research and curriculum development in mathematics education. Nueva York: Springer.

2. Artigue, M., Batanero, C., & Kent, P. (2007). Mathematics thinking and learning at post-secondary level. In F. K. Lester Jr.. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1011–1050). Reston: National Council of Teachers of Mathematics.

3. Beltrán-Meneu, M. J., Murillo-Arcila, M., & Albarracín, L. (2016). Emphasizing visualization and physical applications in the study of eigenvectors and eigenvalues. Teaching Mathematics and its Applications: An International Journal of the IMA, 36(3), 123–135.

4. Blumer, H. (1969). Symbolic interactionism: Perspectives and method. Englewood Cliffs: Prentice-Hall.

5. Bouhjar, K., Andrews-Larson, C., Haider, M., & Zandieh, M. (2018). Examining Students’ Procedural and Conceptual Understanding of Eigenvectors and Eigenvalues in the Context of Inquiry-Oriented Instruction. In S. Stewart, C. Andrews-Larson, A. Berman, & M. Zandieh (Eds.), Challenges In Teaching Linear Algebra Challenges and Strategies in Teaching Linear Algebra (pp. 193–216). Cham: Springer.

6. Çağlayan, G. (2015). Making sense of eigenvalue-eigenvector relationships: Math majors’ linear algebra–geometry connections in a dynamic environment. Journal of Mathematical Behavior, 40, 131–153.

7. Cobb, P., & Yackel, E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. Educational Psychologist, 31, 175–190.

8. Dorier, J.-L. (Ed.). (2000). On the teaching of linear algebra. Dordrecht: Kluwer Academic.

9. Dubinsky, E., & McDonald, M. (2001). APOS: A constructivist theory of learning. In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp. 275–282). Dordrecht: Kluwer Academic.

10. Glaser, B., & Strauss, A. (1967). The discovery of grounded theory: Strategies for qualitative research. Chicago: Aldine Publishing Company.

11. Gol Tabaghi, S., & Sinclair, N. (2013). Using dynamic geometry software to explore eigenvectors: The emergence of dynamic-synthetic-geometric thinking. Technology, Knowledge and Learning, 18(3), 149–164.

12. Harel, G. (2000). Three principles of learning and teaching mathematics. In J.-L. Dorier (Ed.), On the teaching of linear algebra. Dordrecht: Kluwer Academic.

13. Henderson, F., Rasmussen, C., Sweeney, G., Wawro, M., & Zandieh, M. (2010). Symbol sense in linear algebra: A start toward eigen theory. Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education, Raleigh, NC. Retrieved from http://sigmaa.maa.org/rume/crume2010. Accessed 1 Sept 2015.

14. Hiebert, J., & Lafevre, P. (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1–27). Hillsdale: Lawrence Erlbaum Associates.

15. Hillel, J. (2000). Modes of description and the problem of representation in linear algebra. In J.-L. Dorier (Ed.), On the teaching of linear algebra (pp. 191–207). Dordrecht: Kluwer.

16. Larson, C., & Zandieh, M. (2013). Three interpretations of the matrix equation Ax = b. For the Learning of Mathematics, 33(2), 11–17.

17. Lay, D., Lay, S., & McDonald, J. (2016). Linear algebra and its applications (5th edn.). Essex: Pearson Education.

18. Lesh, R., Hoover, M., Hole, B., Kelly, A., & Post, T. (2000). Principles for developing thought-revealing activities for students and teachers. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 591–646). New Jersey: Lawrence Erlbaum.

19. Miles, M. B., Huberman, A. M., & Saldaña, J. (2014). Fundamentals of qualitative data analysis. In Qualitative Data Analysis: A Methods Sourcebook (3rd edn.). Thousand Oaks: Sage Publications Inc.

20. Nyman, M. A., Lapp, D. A., St. John, D., & Berry, J. S. (2010). Those do what? Connecting eigenvectors and eigenvalues to the rest of linear algebra: Using visual enhancements to help students connect eigenvectors to the rest of linear algebra. International Journal for Technology in Mathematics Education, 17(1), 35–41.

21. Plaxco, D., Zandieh, M., & Wawro, M. (2018). Stretch directions and stretch factors: A sequence intended to support guided reinvention of eigenvector and eigenvalue. In S. Stewart, C. Andrews-Larson, A. Berman & M. Zandieh (Eds.), Challenges In Teaching Linear Algebra. ICME-13 Monographs (pp. 175–192). Cham: Springer.

22. Rasmussen, C., & Wawro, M. (2017). Post-calculus research in undergraduate mathematics education. In J. Cai (Ed.), The compendium for research in mathematics education (pp. 551–579). Reston: National Council of Teachers of Mathematics.

23. Salgado, H., & Trigueros, M. (2015). Teaching eigenvalues and eigenvectors using models and APOS Theory. The Journal of Mathematical Behavior, 39, 100–120.

24. Saxe, G. B. (2002). Children’s developing mathematics in collective practices: A framework for analysis. Journal of the Learning Sciences, 11, 275–300.

25. Tall, D. O. (2004). Building theories: The three worlds of mathematics. For the Learning of Mathematics, 24(1), 29–32.

26. Thomas, M. O. J., & Stewart, S. (2011). Eigenvalues and eigenvectors: Embodied, symbolic and formal thinking. Mathematics Education Research Journal, 23(3), 275–296.

27. von Glasersfeld, E. (1995). Radical constructivism: A way of knowing and learning. Bristol: Falmer Press.

28. Watson, K., Wawro, M., Zandieh, M., & Kerrigan, S. (2017). Knowledge about student understanding of eigentheory: Information gained from multiple choice extended assessment. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown (Eds.), Proceedings of the 20th annual conference on research in undergraduate mathematics education (pp. 311–325), San Diego, CA: SIGMAA on RUME.

29. Wawro, M., Watson, K., & Zandieh, M. (2018). Student understanding of linear combinations of eigenvectors. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, and S. Brown (Eds.), Proceedings of the 21st annual conference on research in undergraduate mathematics education (pp. 1372–1378), San Diego, CA: SIGMAA on RUME.

30. Wawro, M., Zandieh, M., Rasmussen, C., & Andrews-Larson, C. (2013). Inquiry oriented linear algebra: Course materials. Retrieved from http://iola.math.vt.edu. Accessed 1 Sept 2015.

31. Wawro, M., Zandieh, M., & Watson, K. (2018). Delineating aspects of understanding eigentheory through assessment development. In V. Durand-Guerrier, R. Hochmuth, S. Goodchild, & N.M. Hogstad (Eds.), Proceedings of INDRUM 2018 Second Conference of the International Network for Didactic Research in University Mathematics (pp. 275–284), Kristiansand: University of Agder and INDRUM.

32. Zandieh, M., Adiredja, A., & Knapp, J. (2018). Exploring Everyday Examples to ExplainBasis from Eight German Male Graduate STEM Students. (Manuscript submitted for publication).

33. Zandieh, M., & Andrews-Larson, C. (2018). Solving linear system: Reconstructing unknowns to interpret row reduced matrices. (Manuscript submitted for publication).

34. Zandieh, M., Plaxco, D., Wawro, M., Rasmussen, C., Milbourne, H., & Czeranko, K. (2015). Extending multiple choice format to document student thinking. In T. Fukawa-Connelly, N. Infante, K. Keene, and M. Zandieh (Eds.), Proceedings of the 18th annual conference on research in undergraduate mathematics education (pp. 1094–1100), Pittsburgh, PA: SIGMAA on RUME.

35. Zandieh, M., Wawro, M., & Rasmussen, C. (2017). An example of inquiry in linear algebra: The roles of symbolizing and brokering. PRIMUS, 27(1), 96–124.

## 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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