Elementary vectors of a real subspace are vectors with a maximal number of zero entries, or, in other words, the sparsest vectors of a subspace.
By applying the sign function component-wise to a real vector, we obtain a sign vector. If we do this for each vector of a real subspace, we obtain a so-called oriented matroid. Many properties, like support or orthogonality, can be motivated for sign vectors in an analogous way.
In this lecture, we learn how to compute with elementary and sign vectors. We discuss algebraic and combinatorial properties of sign vectors and the corresponding algorithmic methods. Sign vectors encode also many properties of related geometric objects, for example, point configurations and solutions of linear inequalities (polytopes). We will also discuss applications of sign vectors to positive solutions of polynomial equations and metabolic networks in systems biology.
In the exercises for this lecture, we will also use the open-source computer algebra system SageMath, in particular, the cloud application CoCalc. No prior experience is needed.
Further information and references will be available at the moodle site for the course.