Formulas for calculating the dimensions of the sums and the intersections of a family of linear subspaces with applications

Abstract

The union and the intersection of subspaces are fundamental operations in geometric algebra, while it is well known that both sum and intersection of linear subspaces in a vector space over a field are linear subspaces as well. A fundamental problem on a given family of linear subspaces is to determine the dimensions of their sum and intersection, as well as the orthogonal projectors onto the sum and the intersection of the family of linear subspaces. In this article, we first revisit an analytical formula for calculating the dimension of the intersection of a family of linear subspaces \({{\mathcal {M}}}_1, \, {{\mathcal {M}}}_2, \ldots , {\mathcal M}_k\) in a finite-dimensional vector space, and establish some analytical expressions of the orthogonal projector onto the intersection using generalized inverses of matrices. We then establish a general formula for calculating the dimension of the sum of the family of linear subspaces. We also give some applications of these formulas in determining the maximum and minimum dimensions of the intersections of column spaces of matrix products involving generalized inverses.