Invariant Subspaces: An Alternative for Introducing Eigenvectors and Eigenvalues

Abstract

The concepts of eigenvalue and eigenvector are typically approached algorithmically in introductory linear algebra courses. However, a more conceptual orientation involves connecting these notions to the concept of one-dimensional invariant subspace, which allows for the introduction of eigenvectors prior to eigenvalues. In this study, we present data collected from interviews with two linear algebra instructors as they worked with a specific linear transformation in both paper-and-pencil and dynamic geometry environments. The data were analyzed using the perspectives of APOS theory and the theory of Mathematical Working Spaces in a complementary manner. The results indicate that dynamic representations facilitate the establishment of relationships between eigenvectors, eigenvalues, and invariant subspaces. This approach proves to have potential for developing a deeper understanding of the related concepts.