RUME 25
2023
Omaha, Nebraska
Synthetic vs. Analytic Formalism of Matrix Representations in Linear Algebra: What it Means to Diagonalize a Linear Transformation
Page: 770
The contributed report highlights linear algebra students’ understandings of the matrix representations of certain linear transformations defined on with respect to given bases in a DGS-MATLAB assisted pedagogical environment. The research focused on the diversity of strategies that students utilized specifically in the process of diagonalizing linear transformations in a series of in-depth qualitative interviews within a framework of modes of description and thinking in linear algebra (Dreyfus, Hillel & Sierpinska, 1998; Sierpinska, 2000). Data analysis revealed three main categories: (i) synthetic-geometric mode manifested as the visual alignment strategy in which students visualized each image vector as a multiple of the corresponding preimage vector; (ii) analytic arithmetic mode prevailed in the process of expressing the image vectors in terms of the preimage vectors; (ii) analytic structural mode was observed in the process of determining the diagonal form of the linear transformation with reference to the transition matrices and the commutative diagram.