For a real $2\times2$ matrix, the column vectors give the images of the coordinate axes. Absolute determinant measures area scaling, determinant sign measures orientation when nonsingular, and rank distinguishes plane, line or point images. Orthogonal matrices preserve lengths and angles; general linear maps do not.
Like how this is organised?
Build your own knowledge tree on knowen.md: notes, decisions and tasks in one connected markdown graph, shared with your team and your AI agents.