Overview

An affine space is a point space associated with a linear space A as in Section 2.5. The dimension of is defined as the dimension of A. The solution of an inhomogeneous linear system is an abstract example of an affine space.

Literature: Berger, Greub, Schaal

9.1 Affine Coordinates

Let a be a fixed point of an affine space and let a ₁, ..., a _n be a basis of the associated linear space A. For simplicity, regard points and vectors as elements of where d ? n. Then, by the properties mentioned in Section 2.5, a point p of has a unique representation

or in compact form p = a+ Ax. The x ₁, ..., x _n are called affine coordinates of p with respect to the affine system a; a ₁, ..., a _n. Let q = a+ Ay denote a second point. Then the vector p - q has the coordinates x - y. Sometimes the notation is used to indicate that has dimension n.

Figure 9.1: Affine system.

Every linear space has the structure of an affine space since the coordinates of a vector can be treated as the coordinates of a point and the zero vector can be interpreted as the origin.

9.2 Affine Subspaces

An affine subspace of