Investigating difficulties and enhancing understanding in linear algebra: Leveraging SageMath and ChatGPT for (orthogonal) diagonalization and singular value decomposition.

Abstract

We explored some common challenges faced by undergraduate students when studying linear algebra, particularly when dealing with algorithmic thinking skills required for topics such as matrix factorization, focusing on (orthogonal) diagonalization and singular value decomposition (SVD). To address these challenges, we introduced SageMath, a Python-based open-source computer algebra system, as a supportive tool for students performing computational tasks despite its static output nature. We further examined the potential of dynamic ChatGPT, an AI-based chatbot, by requesting examples or problem-solving assistance related to (orthogonal) diagonalization or the SVD of a specific matrix. By reinforcing essential concepts in linear algebra and enhancing computational skills through effective practice, mastering these topics can become more accessible while minimizing mistakes. Although static in nature, SageMath proved valuable for confirming calculations and handling tedious computations because of its easy-to-understand syntax and accurate solutions. However, although dynamic ChatGPT may not be fully reliable for solving linear algebra problems, the errors it produces can serve as a valuable resource for improving critical thinking skills.