The aim of this paper is on the one hand to discuss from an APOS (Action-Process-Object-Schema) theory perspective the mental constructions involved in the learning of linear algebra, through examples concerning the linear transformation concept and related notions. On the other hand, methodological issues related to the design of research instruments and implementation of didactic interviews are discussed, supported by empirical data. Detailed analysis of transcripts from an interview with a student focuses on strategies used in interviewing as well as the mental stages involved in the construction of some linear algebra concepts. Due to the strategies employed, during the interview it is possible to witness the transition between different conceptions. A discussion of the relationships and interactions between different mental structures and mechanisms that play a role in the development of knowledge is provided, including theoretical considerations on the matter. Recommendations about pedagogical strategies are included.