This study investigates how college students and first-year graduate students understand the matrix representation of linear transformations, expressed as ( A[v]_E = [L(v)]_F ), where ( A ) is the matrix of the transformation ( L ) with respect to bases ( E ) and ( F ). While many can perform the computations, they often struggle to interpret what the matrix means and how it represents the transformation, especially in abstract vector spaces and nonstandard bases. These challenges reveal not only procedural gaps but also conceptual difficulties.

To examine these issues, the study draws on the Onto-Semiotic Approach (OSA) as its theoretical foundation. OSA has been widely used in mathematics education research and teaching, especially in Europe and South America, over the past three decades. Its growing adoption in the context of college mathematics in the United States reflects a broader interest in frameworks that go beyond correctness and focus on meaning-making. OSA analyzes how students interact with mathematical objects, language, representations, and processes during learning.

To complement this, Statistical Implicative Analysis (SIA), implemented via the CHIC algorithm, was also applied to uncover quasi-implicative relationships among students’ conceptual competencies based on written tasks and interviews. Data were collected from upper-level undergraduates and first-year graduate students in Fall 2024 and Spring 2025.

The analysis identified common obstacles, such as confusion between vectors and their coordinates, difficulty transitioning between symbolic and structural views, and limited understanding of how matrices represent transformations between bases. By integrating OSA and SIA, this research offers insight into how students make sense of linear algebra and suggests strategies for promoting deeper conceptual understanding.