This course introduces the core structures of linear algebra and develops the theoretical and computational tools used throughout mathematics, computer science, and applied sciences.
We begin with vector spaces, studying their structure through subspaces, linear combinations, free families, generating sets, bases, and dimension. Special attention is given to finite-dimensional spaces and decompositions into supplementary subspaces.
Next, we examine linear maps, focusing on their algebraic properties, kernels and images, and the rank theorem, which links dimension with injectivity and surjectivity. Composition and invertibility of linear transformations are also analyzed.
The course then moves to matrices as concrete representations of linear maps. Students learn matrix operations, matrix spaces, determinants, invertibility, and the notion of matrix rank together with its invariance properties.
Finally, we study systems of linear equations, their matrix formulation, rank-based solvability conditions, and solution techniques including Cramer’s rule.
- Teacher: Soufyane Mokhtari