Affine geometry is the geometry of an n-dimensional vector space together with its inhomogeneous linear structure. Accordingly, this chapter covers basic material on linear geometries and linear transformations. The inhomogeneous linear maps that we allow as transformations of affine space include translations such as (x, y) ↦ (x + a, y + b), dilations such as (x, y) ↦ (2x, 2y) and ‘shear’ maps such as (x, y) ↦ (x, x + y). It is impossible to define an origin, distances between points, or angles between lines in a way which makes them invariant under these transformations, or to compare ratios of distances in different directions. However, the line PQ through two points P and Q of An makes perfectly good sense; this is also called the affine span 〈P, Q〉 of P and Q. An affine line is a particular case of an affine linear subspace E ⊂ An; I can view an affine linear subspace as the affine span 〈P1, …, Pk〉 of a finite set of points, or as the set of solutions of a system of inhomogeneous linear equations Mx = b. Arbitrary affine linear maps take affine linear subspaces into one another, and also preserve collinearity of points, parallels and ratios of distances along parallel lines; all of these are thus well defined notions of affine geometry.