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.