### 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 Ax= λx; 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.

Original language | English (US) |
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Journal | ZDM - Mathematics Education |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

### Fingerprint

### Keywords

- Eigenspace
- Linear algebra
- Linear combination
- Student reasoning

### ASJC Scopus subject areas

- Education
- Mathematics(all)

### Cite this

*ZDM - Mathematics Education*. https://doi.org/10.1007/s11858-018-01022-8

**Student understanding of linear combinations of eigenvectors.** / Wawro, Megan; Watson, Kevin; Zandieh, Michelle.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Student understanding of linear combinations of eigenvectors

AU - Wawro, Megan

AU - Watson, Kevin

AU - Zandieh, Michelle

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 Ax= λx; 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.

AB - 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 Ax= λx; 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.

KW - Eigenspace

KW - Linear algebra

KW - Linear combination

KW - Student reasoning

UR - http://www.scopus.com/inward/record.url?scp=85059871140&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059871140&partnerID=8YFLogxK

U2 - 10.1007/s11858-018-01022-8

DO - 10.1007/s11858-018-01022-8

M3 - Article

JO - ZDM - International Journal on Mathematics Education

JF - ZDM - International Journal on Mathematics Education

SN - 1863-9690

ER -