Speaker
Description
The use of real or realistic problems to introduce university students to linear algebra concepts through modeling has proven effective in stimulating student learning. We present an experience based on mathematical modeling and APOS theory (Action, Process, Object, and Schema) to introduce the concepts of matrix transformation and inverse matrix transformation in the context of an introductory Linear Algebra course for engineering students.
The instructional sequence is inspired by Blind Source Separation (BSS), a complex engineering problem concerned with recovering original sources from observed mixtures. Building on a prior in-depth study of BSS models, we identify a linear, noise-free formulation in which observations are modeled as linear combinations of sources. From this formulation, we carried out a transposition of the structural elements of the problem to an introductory linear algebra setting, interpreting signals as vectors and mixing processes as linear transformations.
Based on this model, the inverse transformation emerges as a natural response to the associated inverse problem. Rather than introducing the inverse matrix as a formal object or technique, the proposed genetic decomposition describes how the need for an inverse transformation arises from interpreting linear transformations as models of mixing processes. The construction involves the coordination of various mathematical processes, including the matrix--vector product, systems of linear equations, and the interpretation of linear transformations as functions between vector spaces. Invertibility is conceptualized in terms of information loss and the impossibility of reversing certain transformations.
This work contributes to the literature through the design of a modeling-based instructional situation that foregrounds the emergence and construction of inverse transformations in an introductory linear algebra context.