This exploratory study investigated 10 pre-service mathematics students' understanding of vector subspaces at a university in Zimbabwe. The study involved an in-depth document analysis of the students' written responses to five items based on the vector subspace concept. An APOS (Action--Process--Object--Schema) approach was used to identify the difficulties associated with developing the mental constructions of the vector subspace concept. The responses revealed that most of these pre-service teachers used reasoning associated with Action, or not even Action, conceptions of the prerequisite concepts of binary operations and sets, which hampered their engagement with the higher-level concepts of checking axioms. It is recommended that students should work with different sets and different types of binary operations, allowing them to routinise the Action of carrying out the binary operations on the sets first before moving to abstract applications. The data also revealed that students had difficulties in making links within the embodied, symbolic and formal worlds of thinking. It is suggested that instructors ensure that these links form an explicit focus of tasks.