Formalizing geometry by introducing coordinate systems and using linear algebra captures only a part of the material traditionally regarded as geometry. The missing part has much to do with matters of scale; beyond that, the selection of an origin as a point of reference runs contrary to all the independence of position and orientation which characterizes geometrical reasoning.
Two different, but nevertheless related, concepts recapture geometry within a framework of linear algebra. Including translations along with the linear transformations, and expecially rotations, leads to the concept of an affineaffine space, at the price of working in a nonlinear environment. Projections not only eliminate the nonlinearity; they offer the additional advantages of scaling and inversion - reflections in spheres.