Linear Algebra: a Privileged Context to Develop Abstract Notions Using Spatial Intuition

摘要

The proposed learning situations can be presented either as specific examples or as problems to solve with concrete values. Depending of what we want to achieve, it is possible to explore them with different points of view. We can ask for the parametric equations, the specific points that satisfy the given relation, and so on. The first three learning situations deal with the geometric relations that exist between solutions of system of equations, affine subspaces and vector subspaces. We saw that the cross-product does not exist for space of dimension higher than three. The next three learning situations use the preceding geometric relations and some metric relations to solve specific problems of interpolation with biaffine functionals. The last nine geometric learning situations exploit the Gram-Schmidt orthogonalization of vectors. This tool can effectively replace the use of cross-product to solve some problems. Introducing spatial geometric problems into linear algebra helps students to learn and understand abstract notions. The main objective of the learning situations is to give to the teacher means to illustrate and exploit the theory.