ABSTRACT
In this chapter, the authors provide a short overview of the background that is necessary to make the book mathematically self-contained. In accordance with the formal-geometric approach of Geometric Data Analysis (GDA), abstract linear algebra is given a central place. In GDA, the clouds of points are constructed from numerical tables. The construction is an elaborate process that comprises two phases: formalization and application of mathematical theory. The authors describe the basic matrix operations and review finite-dimensional vector spaces, Euclidean properties and spectral decomposition. They present the basics of multidimensional geometry. The primitive concepts of multidimensional geometry are basically those of elementary geometry, namely points, lines, planes, geometric vectors, etc. Multidimensional geometry enables to carry over both the spatial intuition and the rigorous coordinate-free mode of reasoning proper to pure geometry. The authors outline the fundamental notions of an affine subspace, namely Cartesian frame, and affine map.
