<p>Research on the teaching and learning of linear algebra has been ongoing for several decades, but it has recently been renewed in the context of non-routine tasks and technological ecosystems, including artificial intelligence. This article analyzes two key transitions: (a) from school-level geometry and algebra to linear algebra, and (b) between different paradigms of linear algebra. We propose a fundamental situation in which the goal is to find the vertices of a polygon given its midpoints. This problem can be modelled using systems of linear equations that may have a unique solution, infinitely many, or none. The situation was divided into three tasks and implemented in three universities in the Valparaíso region that train prospective mathematics teachers. Thirty pre-service teachers participated. Data collected included audio recordings, screen captures, and written productions. The analysis was conducted using the categories of the Mathematical Working Space and the paradigms of linear algebra. Results show that most students were able to transition from geometry to linear algebra, mainly within the first paradigm. The mathematical work was predominantly semiotic and instrumental, but a gradual increase in discursive work was observed all along the didactical situation. Various digital tools were used—mainly GeoGebra and MatrixCalculator, with some use of Wolfram Alpha and ChatGPT—serving to both find algebraic solutions and validate them geometrically. The findings suggest that a long sequence of several open tasks involving domain changes and diverse technologies can support the introduction of linear algebra in teacher education.</p>