Teaching and Learning of Linear Algebra

Linear algebra occupies a central position in the mathematical curriculum, bridging abstract theory and practical applications across science, engineering and data science. Teaching and learning in this domain involve developing a coherent understanding of fundamental concepts such as vector spaces, linear transformations, systems of equations, eigenvalues and eigenvectors. Contemporary research emphasises the interplay between procedural fluency—manipulating matrices and executing algorithms—and conceptual understanding, which allows students to interpret results graphically or in application contexts. Educators are exploring diverse pedagogical strategies, including inquiry-based learning, visualisation tools and collaborative problem solving, to address widespread difficulties such as abstraction, symbol manipulation and transfer between representations. Digital platforms and dynamic geometry environments have become instrumental in making the structure of vector spaces and linear mappings visible, while curriculum design studies advocate for integrating modelling tasks early to strengthen students’ higher-order reasoning. Moreover, studies of student misconceptions highlight the necessity of targeted interventions that diagnose and remediate specific gaps in understanding. This global scholarly effort seeks to equip learners with both the technical skills and the conceptual frameworks needed for advanced study and real-world problem solving.

Research from Nature Portfolio

No recent Nature Portfolio content available.

Teaching and Learning of Linear Algebra publication trend

The graph below shows the total number of articles in teaching and learning of linear algebra across all publications each year (not limited to Nature Index journals).

Technical terms

Vector space: A collection of objects called vectors, closed under addition and scalar multiplication, satisfying specific axioms such as associativity and distributivity.

Matrix: A rectangular array of numbers or symbols representing a linear transformation or a system of linear equations.

Linear combination: An expression formed by multiplying vectors by scalars and adding the results.

Span: The set of all linear combinations of a given list of vectors, describing a subspace of a vector space.

Elementary row operations: The three fundamental operations (row swapping, scalar multiplication of a row, and adding a multiple of one row to another) used to solve systems of linear equations via Gaussian elimination.

Linear independence: A property of a set of vectors whereby no nontrivial linear combination of them equals the zero vector, indicating that each vector adds a new dimension to the span.

References

1. Undergraduate students’ conceptualization of elementary row operations in solving systems of linear equations. Eurasia Journal of Mathematics Science and Technology Education (2023).
2. Symbolizing lines and planes as linear combinations in a dynamic geometry environment. The Journal of Mathematical Behavior (2022).
3. Misconceptions and resulting errors displayed by in service teachers in the learning of linear independence. International Electronic Journal of Mathematics Education (2022).