Path Connectedness and Invertible Matrices

Though path-connectedness is a very geometric and visual property, math lets us formalize it and use it to gain geometric insight into spaces that we cannot visualize. In these notes, we will consider spaces of matrices, which (in general) we cannot draw as regions in R or R. To begin studying these spaces, we first explicitly define the concept of a path. Definition 1.1. A path in X is a continuous function φ : [0, 1]→ X.