Dissertation Title:          Undergraduate Mathematics Students' Understanding of and Connections Between Group Homomorphisms and Linear Transformations

Abstract Title:   Many students in a first linear algebra class find difficulty in transitioning from algorithmic matrix computations to the use of formal vector space theory. One of the many difficult concepts at the heart of this formal world is that of the linear transformation. Some of these students continue on to a class in abstract algebra, where they gain more experience with formal mathematics and learn of groups and group homomorphisms - concepts which share a great deal in common with vector spaces and linear transformations. This study explores students' concept images of both linear transformations and group homomorphisms, how these compare to one another, and what connections students make between these and their related concepts across classes. Data for the study comes from qualitative, semi-structured interviews with ten students across three college campuses in Kentucky, as well as three experienced mathematicians at one of these campuses. The data show that students struggled to recall linear transformations by the time they learned of group homomorphisms. Analysis of the data revealed themes among certain subsets of students, such as the understanding of both morphisms as structure-preserving functions, the importance of geometric reasoning for linear transformations, and an understanding of the similarity in the conditions required in the definitions of both morphisms. Connections between related concepts in  abstract algebra and linear algebra were often indicated by way of analogy. The data show students at all levels of analogical reasoning, from an already integrated schema, to those still actively making some connections, to students who struggled to make any analogies across the curriculum. Each level is discussed through case analyses of key participants to display the varied ways in which students both succeed and struggle with making these connections.